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Types of Numbers

Classification of Numbers

Every number you encounter in mathematics — from the price of a bus ticket to the digits of π — belongs to one or more carefully defined families, and knowing these families is the foundation of every quantitative question in SSC CGL and similar competitive exams.

Definition: A number system is a structured classification of numbers into sets based on their properties, such as whether they can be expressed as a ratio, whether they have a repeating decimal expansion, and how they behave under arithmetic operations.

Natural Numbers (N)

Definition: The set of natural numbers is N = {1, 2, 3, 4, 5, …}. These are the counting numbers — the ones a child first learns to count objects with.

Key properties:

Smallest natural number: 1
No upper bound — they continue infinitely
Closed under addition and multiplication (adding or multiplying two natural numbers always gives a natural number)
NOT closed under subtraction (3 − 5 = −2, which is not in N)

Why it matters: Every counting operation — ranking candidates in an exam, counting factories in a district — involves natural numbers.

Whole Numbers (W)

Definition: W = {0, 1, 2, 3, 4, …} — natural numbers plus zero.

Smallest whole number: 0
Zero is the additive identity: a + 0 = a for any whole number a
Every natural number is also a whole number, but 0 is not a natural number

Common misconception: Many students treat "natural numbers" and "whole numbers" as synonyms. They are not — the difference is the presence of zero.

Integers (Z)

Definition: Z = {…, −3, −2, −1, 0, 1, 2, 3, …} — all positive whole numbers, zero, and all negative whole numbers.

Integers are closed under addition, subtraction, and multiplication
Division of two integers need not be an integer (7 ÷ 2 = 3.5)
The letter Z comes from the German word Zahlen (numbers)

Real-world example: Temperature in Shimla in January can be −5°C, −3°C, or 2°C — all integers.

Rational Numbers (Q)

Definition: A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Examples: 3/4, −7/2, 0.5 (= 1/2), 0.333… (= 1/3), 6 (= 6/1)

Key insight — decimal behaviour:

A rational number always has either a terminating decimal (e.g., 1/4 = 0.25) or a recurring (repeating) decimal (e.g., 1/3 = 0.333…)
If you see a decimal that terminates or repeats, it is rational

Why it matters: Fractions appear in ratio, proportion, percentage, and speed-distance problems — all high-frequency SSC topics.

Irrational Numbers

Definition: An irrational number cannot be expressed as p/q for any integers p and q. Its decimal expansion is non-terminating and non-repeating.

Common irrational numbers to memorise:

√2 ≈ 1.41421…
√3 ≈ 1.73205…
√5 ≈ 2.23606…
π ≈ 3.14159…
e ≈ 2.71828…

Common misconception: Students often think 22/7 = π. It does NOT — 22/7 is a rational approximation; π itself is irrational.

Another trap: √4 = 2, which IS rational. Not every square root is irrational — only the square root of a non-perfect-square is irrational.

Real Numbers (R)

Definition: The set of real numbers is the union of all rational and irrational numbers. Every point on the number line corresponds to exactly one real number.

⚖️ Subsets of Real Numbers

Set	Notation	Contains	Example
Natural	N	{1, 2, 3, …}	7
Whole	W	{0, 1, 2, 3, …}	0
Integer	Z	{…−2,−1, 0, 1, 2…}	−5
Rational	Q	p/q, q≠0	3/4
Irrational	—	Non-repeating decimals	√2
Real	R	Q ∪ Irrational	All of the above

The hierarchy: N ⊂ W ⊂ Z ⊂ Q ⊂ R

Prime Numbers

Definition: A prime number has exactly two distinct positive factors: 1 and itself.

Important facts:

2 is the only even prime number
1 is NOT prime (it has only one factor: itself)
The number of primes below 100 is 25

List of all 25 primes below 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Why it matters: Prime factorisation is the engine behind LCM, HCF, number of divisors, and many SSC number theory problems.

Composite Numbers

Definition: A composite number has more than two factors — i.e., it can be divided by at least one number other than 1 and itself.

Examples: 4 (factors: 1, 2, 4), 12 (factors: 1, 2, 3, 4, 6, 12)

Key fact: Every composite number can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic).

Example: 360 = 2³ × 3² × 5

Special case: 1 is neither prime nor composite. This is one of the most frequently tested tricky facts in SSC CGL.

Co-prime Numbers

Definition: Two numbers are co-prime (or relatively prime) if their HCF = 1 — they share no common factor other than 1.

Examples:

8 and 15: factors of 8 = {1, 2, 4, 8}; factors of 15 = {1, 3, 5, 15}; HCF = 1 → co-prime ✓
4 and 9: HCF = 1 → co-prime ✓
6 and 9: HCF = 3 → NOT co-prime ✗

Key insight: Co-prime numbers need not be prime themselves — 8 and 15 are co-prime even though neither is prime.

Why it matters: Co-primality appears in LCM/HCF questions and in fraction simplification problems.

Even and Odd Numbers

Even numbers: Divisible by 2. Last digit is 0, 2, 4, 6, or 8.
Odd numbers: Not divisible by 2. Last digit is 1, 3, 5, 7, or 9.

Arithmetic rules (heavily tested in SSC):

Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even

Perfect Numbers

Definition: A perfect number equals the sum of all its proper divisors (all divisors except itself).

Example: 6 → proper divisors: 1, 2, 3 → sum = 6 ✓
Example: 28 → proper divisors: 1, 2, 4, 7, 14 → sum = 28 ✓

The next perfect number after 28 is 496. For SSC purposes, just remember 6 and 28.

Worked Example — Classifying a Number

Question: Classify the number 0.171717… and determine which sets it belongs to.

Solution:

Step 1: The decimal 0.1717… has a repeating block "17". A non-terminating but repeating decimal is always rational.

Step 2: Express as fraction: Let x = 0.1717…
Then 100x = 17.1717…
100x − x = 17 → 99x = 17 → x = 17/99

Step 3: 17/99 is in the form p/q (p=17, q=99, q≠0).

Conclusion: 0.1717… is a rational number, and therefore also a real number. It is not a natural number, whole number, or integer.

🔑 Key points

✓- Natural numbers start at 1; whole numbers start at 0; integers include negatives
✓- Every natural ⊂ whole ⊂ integer ⊂ rational ⊂ real
✓- Rational decimals terminate or repeat; irrational decimals never repeat
✓- 2 is the only even prime number
✓- 1 is neither prime nor composite — this is a favourite SSC trap
✓- Co-prime means HCF = 1, not that either number is prime
✓- There are exactly 25 prime numbers less than 100
✓- Perfect numbers: 6, 28 (sum of proper divisors equals the number)

🧠 Memory hook

"Never Will I Quit Reaching" → Natural, Whole, Integer, Rational, Real — each set contains the previous one. For primes: "2 is the lone ranger" — the only even prime.

🔁 Recap

✓- Numbers split into Natural → Whole → Integer → Rational → Real (each is a subset of the next)
✓- Rational numbers have terminating or recurring decimals; irrational numbers have neither
✓- Prime: exactly 2 factors; Composite: more than 2; 1: neither
✓- Co-prime pairs have HCF = 1 (e.g., 8 and 15)
✓- Even/odd arithmetic rules are directly tested in SSC CGL
✓- 25 primes exist below 100; memorise the full list

Key Properties of Special Numbers

Certain numbers have such special mathematical properties that they appear as traps and shortcuts in competitive exams again and again — mastering them means converting hard questions into one-step recalls.

Prime Numbers up to 100

Every serious SSC CGL aspirant must have the complete list of 25 primes below 100 memorised:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97

Why memorise the full list? SSC CGL regularly asks "which of the following is prime?" with options like 91, 97, 87, 93 — if you know the list, the answer is instant.

Grouping trick for memory:

Teens primes: 11, 13, 17, 19 (all four — skip 15)
Twenties primes: only 23 and 29
Thirties primes: 31, 37 (only two)
Forties primes: 41, 43, 47 (three)
Fifties primes: 53, 59 (only two — skip 51=3×17, 55=5×11, 57=3×19)
Sixties primes: 61, 67 (only two — 63=9×7, 65=5×13)
Seventies primes: 71, 73, 79 (three)
Eighties primes: 83, 89 (only two — 81=3⁴, 87=3×29)
Ninety primes: 97 only (91=7×13 is a classic trap!)

How to Test if a Number is Prime

Definition: To check if n is prime, test divisibility by all prime numbers up to √n.

Why only up to √n? If n has a factor larger than √n, then its corresponding paired factor must be smaller than √n — so you would have found it already.

Worked illustration:

Test n = 221. √221 ≈ 14.8. Check primes: 2, 3, 5, 7, 11, 13.
- 221 ÷ 13 = 17. So 221 = 13 × 17. Composite.
Test n = 233. √233 ≈ 15.3. Check 2, 3, 5, 7, 11, 13: none divide 233. Prime.

Perfect Numbers

Definition: A perfect number is a positive integer that equals the sum of all its proper divisors (all positive divisors excluding the number itself).

The first two perfect numbers (and the only ones relevant for SSC CGL):

Number	Proper Divisors	Sum
6	1, 2, 3	6 ✓
28	1, 2, 4, 7, 14	28 ✓

The next perfect number is 496, then 8128 — far beyond competitive exam scope.

Why it matters: "Which of the following is a perfect number?" is a direct recall question.

Common misconception: Students sometimes confuse "perfect number" with "perfect square." A perfect square is n² for integer n (4, 9, 16, 25…). These are entirely different concepts.

Twin Primes

Definition: Twin primes are pairs of prime numbers that differ by exactly 2.

Known twin prime pairs (up to 100):

(3, 5)
(5, 7)
(11, 13)
(17, 19)
(29, 31)
(41, 43)
(59, 61)
(71, 73)

Note: (2, 3) differ by 1, not 2 — not twin primes.

Twin Prime Conjecture: It is believed (but not proven) that there are infinitely many twin prime pairs. This is one of the great unsolved problems in mathematics.

Even and Odd Number Properties

These arithmetic rules are tested both directly and embedded in larger problems:

⚖️ Even-Odd Arithmetic Rules

Operation	Result	Example
Even + Even	Even	4 + 6 = 10
Odd + Odd	Even	3 + 7 = 10
Even + Odd	Odd	4 + 3 = 7
Even × Even	Even	4 × 6 = 24
Odd × Odd	Odd	3 × 7 = 21
Even × Odd	Even	4 × 3 = 12
Even − Even	Even	8 − 4 = 4
Odd − Odd	Even	7 − 3 = 4

Key insight for SSC: The product of ANY set of numbers is even if at least one of them is even. The product is odd only if all numbers in the product are odd.

Real-world example: If you buy 3 items priced at ₹5, ₹7, and ₹9 each (all odd), total = ₹21 (odd). Add one item at ₹4 (even) and the product of prices is now even.

Properties of Powers (Even/Odd Exponents)

For SSC number theory and remainder problems:

Even number raised to any positive power → Even
Odd number raised to any positive power → Odd
The last digit of a power follows a cycle (tested in cyclicity problems)

Last digit cycles for common bases:

2: cycle 2, 4, 8, 6 (period 4)
3: cycle 3, 9, 7, 1 (period 4)
7: cycle 7, 9, 3, 1 (period 4)
8: cycle 8, 4, 2, 6 (period 4)
1, 5, 6: last digit never changes (always 1, 5, 6 respectively)

Composite Number Properties

Every composite number n can be written as a product of primes in exactly one way (up to order) — this is the Fundamental Theorem of Arithmetic.

Number of divisors formula: If n = p₁^a × p₂^b × p₃^c …, then number of divisors = (a+1)(b+1)(c+1)…

Example: 360 = 2³ × 3² × 5¹
Number of divisors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24

Worked Example — Applying Multiple Properties

Question: The product of two prime numbers is 77. What is their sum?

Solution:

Step 1: Express 77 as a product. 77 = 7 × 11.

Step 2: Check: 7 is prime (not divisible by 2, 3, 5; √7 < 3). 11 is prime.

Step 3: Sum = 7 + 11 = 18.

Question: How many prime numbers are there between 50 and 70?

Solution:

Step 1: List candidates: 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69.

Step 2: Eliminate composites: 51=3×17, 52=2×26, 54=2×27, 55=5×11, 56=2×28, 57=3×19, 58=2×29, 60=2×30, 62=2×31, 63=7×9, 64=2⁶, 65=5×13, 66=2×33, 68=2×34, 69=3×23.

Step 3: Remaining primes: 53, 59, 61, 67 → 4 primes.

🔑 Key points

✓- There are 25 primes below 100; 91=7×13 is a notorious non-prime trap
✓- To test primality of n, check divisibility by primes up to √n only
✓- Perfect numbers: 6 and 28 (proper divisors sum to the number)
✓- Twin primes differ by exactly 2; pairs up to 100: (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73)
✓- Product of numbers is odd ONLY IF every factor is odd
✓- Sum of two odd numbers is always even
✓- Number of divisors of n = p^a × q^b is (a+1)(b+1)

🧠 Memory hook

"Perfect: Six and Twenty-Eight" — 6 = 1+2+3; 28 = 1+2+4+7+14. For twin primes: think of prime siblings sitting next to each other on the number line with one seat gap.

🔁 Recap

✓- Full prime list below 100 has exactly 25 numbers; memorise by decade groups
✓- Primality test: divide by primes ≤ √n; if none divide, it is prime
✓- Perfect number = sum of proper divisors; 6 and 28 are the exam-relevant ones
✓- Twin primes are prime pairs with difference 2
✓- Product is even if any factor is even; product is odd only if all factors are odd
✓- Number of divisors uses the prime factorisation exponent formula (a+1)(b+1)…

Solved Example: Identifying Number Types

The fastest route to a correct answer in SSC CGL is not just knowing the definition of a prime number — it's being able to identify one (or rule it out) in under 30 seconds without a calculator.

The Core Technique: Square Root Boundary Test

Definition: A number n is prime if and only if it has no prime factor less than or equal to √n.

This works because:

If n = a × b and both a > √n and b > √n, then a × b > n — contradiction.
So at least one factor must be ≤ √n.
Therefore, if no prime ≤ √n divides n, then n has no factor at all (other than 1 and n), so it is prime.

Practical rule:

For numbers up to 100: check divisibility by 2, 3, 5, 7 (since √100 = 10, and 7 is the largest prime below 10)
For numbers up to 200: also check 11 and 13 (since √200 ≈ 14.1)
For numbers up to 400: also check 17 and 19 (since √400 = 20)

Divisibility Quick-Checks

Before doing division, use divisibility rules to eliminate candidates instantly:

Divisor	Rule	Example
2	Last digit even	84: last digit 4 → divisible
3	Sum of digits divisible by 3	93: 9+3=12 → divisible
5	Last digit 0 or 5	95: last digit 5 → divisible
7	No simple rule; just divide
11	(Sum of odd-position digits) − (Sum of even-position digits) = 0 or 11	121: (1+1)−2=0 → divisible
13	No simple rule; just divide

Solved Example 1 (from the lesson stub)

Question: Which of the following is a prime number — 91, 97, 87, 93?

Solution:

Step 1: Examine 91. √91 ≈ 9.5 → check primes 2, 3, 5, 7.

Is 91 even? No.
Digit sum = 9+1 = 10 → not divisible by 3.
Last digit not 0 or 5 → not divisible by 5.
91 ÷ 7 = 13. Yes! So 91 = 7 × 13. Composite.

Step 2: Examine 87. √87 ≈ 9.3 → check 2, 3, 5, 7.

Digit sum = 8+7 = 15 → divisible by 3. 87 = 3 × 29. Composite.

Step 3: Examine 93. √93 ≈ 9.6 → check 2, 3, 5, 7.

Digit sum = 9+3 = 12 → divisible by 3. 93 = 3 × 31. Composite.

Step 4: Examine 97. √97 ≈ 9.8 → check 2, 3, 5, 7.

Not even. Digit sum = 9+7 = 16 → not divisible by 3. Last digit ≠ 0,5. 97 ÷ 7 = 13.857… Not divisible.
No prime ≤ √97 divides 97. 97 is prime.

Conclusion: 97 is the prime.

Why 91 is the classic trap: 91 looks prime because it is not divisible by 2, 3, or 5. But 7 × 13 = 91 — always check 7 for two-digit numbers!

Solved Example 2: Identifying the Type of a Given Number

Question: Classify: 1, 2, 4, 15, 17, 28, 36.

Solution:

Number	Factors	Classification
1	1 only	Neither prime nor composite
2	1, 2	Prime (only even prime)
4	1, 2, 4	Composite
15	1, 3, 5, 15	Composite
17	1, 17	Prime
28	1, 2, 4, 7, 14, 28	Composite; also a perfect number
36	1,2,3,4,6,9,12,18,36	Composite; perfect square (6²)

Solved Example 3: How Many Primes Between Two Numbers?

Question: How many prime numbers lie between 80 and 100?

Solution:

Step 1: Candidates: 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.

Step 2: Eliminate composites quickly —

Even numbers: 82, 84, 86, 88, 90, 92, 94, 96, 98 → all composite
Divisible by 5: 85 (=5×17), 90, 95 (=5×19) → composite
Divisible by 3: 81 (=3⁴), 87 (=3×29), 93 (=3×31), 99 (=9×11) → composite
91 = 7 × 13 → composite

Step 3: Remaining: 83 and 89 and 97.

83: √83 ≈ 9.1. Not divisible by 2, 3, 5, 7. → Prime
89: √89 ≈ 9.4. Not divisible by 2, 3, 5, 7. → Prime
97: already confirmed prime above.

Conclusion: 3 primes between 80 and 100 (83, 89, 97).

Common SSC Traps

Trap 1 — 1 is prime: FALSE. 1 is neither prime nor composite.

Trap 2 — 91 is prime: FALSE. 91 = 7 × 13.

Trap 3 — 2 is not prime (it's even): FALSE. 2 is the only even prime.

Trap 4 — All odd numbers are prime: FALSE. 9 = 3×3, 15 = 3×5, 21 = 3×7 are all odd composites.

Trap 5 — √n must be an integer to be relevant: FALSE. Even if √n is not an integer (e.g., √91 ≈ 9.5), you still check all primes up to the floor of √n.

Worked Shortcut: Composite Identification by Digit Sum

Question: Without dividing, determine whether 561 is prime.

Solution:

Step 1: Digit sum = 5+6+1 = 12. Divisible by 3 → 561 is divisible by 3. Composite. (561 = 3 × 187)

This digit-sum trick eliminates most large numbers before you even attempt division.

🔑 Key points

✓- To test primality of n, check only primes ≤ √n — once you pass √n, stop
✓- For numbers 1–100: checking 2, 3, 5, 7 is sufficient (√100 = 10)
✓- 91 = 7 × 13 is the most common prime-impersonator in SSC exams
✓- 1 is neither prime nor composite — a guaranteed trick question
✓- Use divisibility rules (digit sum for 3; last digit for 2, 5) before attempting division
✓- Between 80 and 100 there are exactly 3 primes: 83, 89, 97
✓- Perfect numbers (6, 28) and twin primes are separate special categories

🧠 Memory hook

"Seven Traps Ninety-One" — whenever you see 91, immediately check ÷7. For the test: "Check up to the Square Root, not a step more."

🔁 Recap

✓- Primality test: find √n, then test divisibility by each prime up to that bound
✓- Apply divisibility rules first (2, 3, 5) to quickly eliminate most candidates
✓- 91 = 7×13, 87 = 3×29, 93 = 3×31 — all favourite composite traps in MCQ options
✓- 97 is the largest prime below 100
✓- 1 is not prime; 2 is prime and is the only even prime
✓- Between any two consecutive perfect squares, count primes carefully using the method above

⚡ Speed Tricks & Shortcuts

💡 Speed tricks & shortcuts

To test if n is prime, only trial-divide by primes ≤ √n (e.g. for 149, check up to 12).
2 is the only even prime; 1 is neither prime nor composite.
Two numbers are co-prime if their HCF = 1 (they needn't be prime themselves, e.g. 8 & 9).
Product of n consecutive integers is always divisible by n!.

⚠️ Common mistake

Don't call 1 a prime and don't forget 2 — "smallest prime is 2, smallest odd prime is 3" is a classic trap.

Types of Numbers — Revision Notes

Quick-revision notes for Types of Numbers — the must-know points for SSC CGL Tier-I/II.

Natural numbers (1,2,3…), Whole numbers (0,1,2…), Integers (…−2,−1,0,1,2…), and their subsets.
Rational numbers can be written as p/q (q≠0); irrational numbers (like √2, π) cannot.
Prime numbers have exactly two factors (1 and itself); 2 is the only even prime; 1 is neither prime nor composite.
Composite numbers have more than two factors; co-prime numbers share HCF 1.
Even numbers are divisible by 2; odd are not. Perfect numbers equal the sum of their proper divisors (e.g. 6 = 1+2+3).
Real numbers = rationals + irrationals; they cover the entire number line.

Types of Numbers — Flashcards (SSC CGL)

Cover the answer, recall, then check. 7 cards on the must-know Types of Numbers facts for SSC CGL.

Q1. What is the only even prime number?
A1. 2.

Q2. Is 1 a prime number?
A2. No — 1 is neither prime nor composite.

Q3. Define a rational number.
A3. A number expressible as p/q where p and q are integers and q ≠ 0.

Q4. Give an example of an irrational number.
A4. √2 (or π) — it cannot be written as a ratio of two integers.

Q5. What are co-prime numbers?
A5. Two numbers whose HCF is 1 (e.g. 8 and 9).

Q6. How many factors does a prime number have?
A6. Exactly two — 1 and the number itself.

Q7. Why is 6 called a perfect number?
A7. Because the sum of its proper divisors (1+2+3) equals 6.

Types of Numbers — Formula Sheet

Key formulas

Classification: natural ⊂ whole ⊂ integer ⊂ rational ⊂ real; irrationals are non-terminating non-repeating.
Prime: exactly two factors; composite: more than two; 1 is neither.
Co-prime: HCF = 1. Perfect number = sum of proper divisors (e.g. 6, 28).
Sum of first n naturals n(n+1)/2; first n odd numbers = n²; first n even = n(n+1).
Rational number lies between any two reals; between a and b: (a+b)/2.
A number is rational iff its decimal terminates or repeats.

🔑 Key points

✓- Prime has exactly 2 factors; 1 is neither prime nor composite.
✓- Co-prime ⇒ HCF = 1.
✓- Sum of first n odd numbers = n².
✓- Rational ⇔ terminating or repeating decimal.

Usage: identify the number type first — it fixes which divisibility/sum rules apply.

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Types of Numbers

Classification of Numbers

Natural Numbers (N)

Definition: The set of natural numbers is N = {1, 2, 3, 4, 5, …}. These are the counting numbers — the ones a child first learns to count objects with.

Key properties:

Smallest natural number: 1
No upper bound — they continue infinitely
Closed under addition and multiplication (adding or multiplying two natural numbers always gives a natural number)
NOT closed under subtraction (3 − 5 = −2, which is not in N)

Why it matters: Every counting operation — ranking candidates in an exam, counting factories in a district — involves natural numbers.

Whole Numbers (W)

Definition: W = {0, 1, 2, 3, 4, …} — natural numbers plus zero.

Smallest whole number: 0
Zero is the additive identity: a + 0 = a for any whole number a
Every natural number is also a whole number, but 0 is not a natural number

Common misconception: Many students treat "natural numbers" and "whole numbers" as synonyms. They are not — the difference is the presence of zero.

Integers (Z)

Definition: Z = {…, −3, −2, −1, 0, 1, 2, 3, …} — all positive whole numbers, zero, and all negative whole numbers.

Integers are closed under addition, subtraction, and multiplication
Division of two integers need not be an integer (7 ÷ 2 = 3.5)
The letter Z comes from the German word Zahlen (numbers)

Real-world example: Temperature in Shimla in January can be −5°C, −3°C, or 2°C — all integers.

Rational Numbers (Q)

Definition: A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0.

Examples: 3/4, −7/2, 0.5 (= 1/2), 0.333… (= 1/3), 6 (= 6/1)

Key insight — decimal behaviour:

A rational number always has either a terminating decimal (e.g., 1/4 = 0.25) or a recurring (repeating) decimal (e.g., 1/3 = 0.333…)
If you see a decimal that terminates or repeats, it is rational

Why it matters: Fractions appear in ratio, proportion, percentage, and speed-distance problems — all high-frequency SSC topics.

Irrational Numbers

Definition: An irrational number cannot be expressed as p/q for any integers p and q. Its decimal expansion is non-terminating and non-repeating.

Common irrational numbers to memorise:

√2 ≈ 1.41421…
√3 ≈ 1.73205…
√5 ≈ 2.23606…
π ≈ 3.14159…
e ≈ 2.71828…

Common misconception: Students often think 22/7 = π. It does NOT — 22/7 is a rational approximation; π itself is irrational.

Another trap: √4 = 2, which IS rational. Not every square root is irrational — only the square root of a non-perfect-square is irrational.

Real Numbers (R)

Definition: The set of real numbers is the union of all rational and irrational numbers. Every point on the number line corresponds to exactly one real number.

⚖️ Subsets of Real Numbers

Set	Notation	Contains	Example
Natural	N	{1, 2, 3, …}	7
Whole	W	{0, 1, 2, 3, …}	0
Integer	Z	{…−2,−1, 0, 1, 2…}	−5
Rational	Q	p/q, q≠0	3/4
Irrational	—	Non-repeating decimals	√2
Real	R	Q ∪ Irrational	All of the above

The hierarchy: N ⊂ W ⊂ Z ⊂ Q ⊂ R

Prime Numbers

Definition: A prime number has exactly two distinct positive factors: 1 and itself.

Important facts:

2 is the only even prime number
1 is NOT prime (it has only one factor: itself)
The number of primes below 100 is 25

List of all 25 primes below 100:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Why it matters: Prime factorisation is the engine behind LCM, HCF, number of divisors, and many SSC number theory problems.

Composite Numbers

Definition: A composite number has more than two factors — i.e., it can be divided by at least one number other than 1 and itself.

Examples: 4 (factors: 1, 2, 4), 12 (factors: 1, 2, 3, 4, 6, 12)

Key fact: Every composite number can be expressed as a unique product of primes (Fundamental Theorem of Arithmetic).

Example: 360 = 2³ × 3² × 5

Special case: 1 is neither prime nor composite. This is one of the most frequently tested tricky facts in SSC CGL.

Co-prime Numbers

Definition: Two numbers are co-prime (or relatively prime) if their HCF = 1 — they share no common factor other than 1.

Examples:

8 and 15: factors of 8 = {1, 2, 4, 8}; factors of 15 = {1, 3, 5, 15}; HCF = 1 → co-prime ✓
4 and 9: HCF = 1 → co-prime ✓
6 and 9: HCF = 3 → NOT co-prime ✗

Key insight: Co-prime numbers need not be prime themselves — 8 and 15 are co-prime even though neither is prime.

Why it matters: Co-primality appears in LCM/HCF questions and in fraction simplification problems.

Even and Odd Numbers

Even numbers: Divisible by 2. Last digit is 0, 2, 4, 6, or 8.
Odd numbers: Not divisible by 2. Last digit is 1, 3, 5, 7, or 9.

Arithmetic rules (heavily tested in SSC):

Even + Even = Even
Odd + Odd = Even
Even + Odd = Odd
Even × Even = Even
Odd × Odd = Odd
Even × Odd = Even

Perfect Numbers

Definition: A perfect number equals the sum of all its proper divisors (all divisors except itself).

Example: 6 → proper divisors: 1, 2, 3 → sum = 6 ✓
Example: 28 → proper divisors: 1, 2, 4, 7, 14 → sum = 28 ✓

The next perfect number after 28 is 496. For SSC purposes, just remember 6 and 28.

Worked Example — Classifying a Number

Question: Classify the number 0.171717… and determine which sets it belongs to.

Solution:

Step 1: The decimal 0.1717… has a repeating block "17". A non-terminating but repeating decimal is always rational.

Step 2: Express as fraction: Let x = 0.1717…
Then 100x = 17.1717…
100x − x = 17 → 99x = 17 → x = 17/99

Step 3: 17/99 is in the form p/q (p=17, q=99, q≠0).

Conclusion: 0.1717… is a rational number, and therefore also a real number. It is not a natural number, whole number, or integer.

🔑 Key points

✓- Natural numbers start at 1; whole numbers start at 0; integers include negatives
✓- Every natural ⊂ whole ⊂ integer ⊂ rational ⊂ real
✓- Rational decimals terminate or repeat; irrational decimals never repeat
✓- 2 is the only even prime number
✓- 1 is neither prime nor composite — this is a favourite SSC trap
✓- Co-prime means HCF = 1, not that either number is prime
✓- There are exactly 25 prime numbers less than 100
✓- Perfect numbers: 6, 28 (sum of proper divisors equals the number)

🧠 Memory hook

"Never Will I Quit Reaching" → Natural, Whole, Integer, Rational, Real — each set contains the previous one. For primes: "2 is the lone ranger" — the only even prime.

🔁 Recap

✓- Numbers split into Natural → Whole → Integer → Rational → Real (each is a subset of the next)
✓- Rational numbers have terminating or recurring decimals; irrational numbers have neither
✓- Prime: exactly 2 factors; Composite: more than 2; 1: neither
✓- Co-prime pairs have HCF = 1 (e.g., 8 and 15)
✓- Even/odd arithmetic rules are directly tested in SSC CGL
✓- 25 primes exist below 100; memorise the full list

Key Properties of Special Numbers

Prime Numbers up to 100

Every serious SSC CGL aspirant must have the complete list of 25 primes below 100 memorised:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97

Why memorise the full list? SSC CGL regularly asks "which of the following is prime?" with options like 91, 97, 87, 93 — if you know the list, the answer is instant.

Grouping trick for memory:

Teens primes: 11, 13, 17, 19 (all four — skip 15)
Twenties primes: only 23 and 29
Thirties primes: 31, 37 (only two)
Forties primes: 41, 43, 47 (three)
Fifties primes: 53, 59 (only two — skip 51=3×17, 55=5×11, 57=3×19)
Sixties primes: 61, 67 (only two — 63=9×7, 65=5×13)
Seventies primes: 71, 73, 79 (three)
Eighties primes: 83, 89 (only two — 81=3⁴, 87=3×29)
Ninety primes: 97 only (91=7×13 is a classic trap!)

How to Test if a Number is Prime

Definition: To check if n is prime, test divisibility by all prime numbers up to √n.

Why only up to √n? If n has a factor larger than √n, then its corresponding paired factor must be smaller than √n — so you would have found it already.

Worked illustration:

Test n = 221. √221 ≈ 14.8. Check primes: 2, 3, 5, 7, 11, 13.
- 221 ÷ 13 = 17. So 221 = 13 × 17. Composite.
Test n = 233. √233 ≈ 15.3. Check 2, 3, 5, 7, 11, 13: none divide 233. Prime.

Perfect Numbers

Definition: A perfect number is a positive integer that equals the sum of all its proper divisors (all positive divisors excluding the number itself).

The first two perfect numbers (and the only ones relevant for SSC CGL):

Number	Proper Divisors	Sum
6	1, 2, 3	6 ✓
28	1, 2, 4, 7, 14	28 ✓

The next perfect number is 496, then 8128 — far beyond competitive exam scope.

Why it matters: "Which of the following is a perfect number?" is a direct recall question.

Common misconception: Students sometimes confuse "perfect number" with "perfect square." A perfect square is n² for integer n (4, 9, 16, 25…). These are entirely different concepts.

Twin Primes

Definition: Twin primes are pairs of prime numbers that differ by exactly 2.

Known twin prime pairs (up to 100):

(3, 5)
(5, 7)
(11, 13)
(17, 19)
(29, 31)
(41, 43)
(59, 61)
(71, 73)

Note: (2, 3) differ by 1, not 2 — not twin primes.

Twin Prime Conjecture: It is believed (but not proven) that there are infinitely many twin prime pairs. This is one of the great unsolved problems in mathematics.

Even and Odd Number Properties

These arithmetic rules are tested both directly and embedded in larger problems:

⚖️ Even-Odd Arithmetic Rules

Operation	Result	Example
Even + Even	Even	4 + 6 = 10
Odd + Odd	Even	3 + 7 = 10
Even + Odd	Odd	4 + 3 = 7
Even × Even	Even	4 × 6 = 24
Odd × Odd	Odd	3 × 7 = 21
Even × Odd	Even	4 × 3 = 12
Even − Even	Even	8 − 4 = 4
Odd − Odd	Even	7 − 3 = 4

Key insight for SSC: The product of ANY set of numbers is even if at least one of them is even. The product is odd only if all numbers in the product are odd.

Real-world example: If you buy 3 items priced at ₹5, ₹7, and ₹9 each (all odd), total = ₹21 (odd). Add one item at ₹4 (even) and the product of prices is now even.

Properties of Powers (Even/Odd Exponents)

For SSC number theory and remainder problems:

Even number raised to any positive power → Even
Odd number raised to any positive power → Odd
The last digit of a power follows a cycle (tested in cyclicity problems)

Last digit cycles for common bases:

2: cycle 2, 4, 8, 6 (period 4)
3: cycle 3, 9, 7, 1 (period 4)
7: cycle 7, 9, 3, 1 (period 4)
8: cycle 8, 4, 2, 6 (period 4)
1, 5, 6: last digit never changes (always 1, 5, 6 respectively)

Composite Number Properties

Every composite number n can be written as a product of primes in exactly one way (up to order) — this is the Fundamental Theorem of Arithmetic.

Number of divisors formula: If n = p₁^a × p₂^b × p₃^c …, then number of divisors = (a+1)(b+1)(c+1)…

Example: 360 = 2³ × 3² × 5¹
Number of divisors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24

Worked Example — Applying Multiple Properties

Question: The product of two prime numbers is 77. What is their sum?

Solution:

Step 1: Express 77 as a product. 77 = 7 × 11.

Step 2: Check: 7 is prime (not divisible by 2, 3, 5; √7 < 3). 11 is prime.

Step 3: Sum = 7 + 11 = 18.

Question: How many prime numbers are there between 50 and 70?

Solution:

Step 1: List candidates: 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69.

Step 2: Eliminate composites: 51=3×17, 52=2×26, 54=2×27, 55=5×11, 56=2×28, 57=3×19, 58=2×29, 60=2×30, 62=2×31, 63=7×9, 64=2⁶, 65=5×13, 66=2×33, 68=2×34, 69=3×23.

Step 3: Remaining primes: 53, 59, 61, 67 → 4 primes.

🔑 Key points

✓- There are 25 primes below 100; 91=7×13 is a notorious non-prime trap
✓- To test primality of n, check divisibility by primes up to √n only
✓- Perfect numbers: 6 and 28 (proper divisors sum to the number)
✓- Twin primes differ by exactly 2; pairs up to 100: (3,5),(5,7),(11,13),(17,19),(29,31),(41,43),(59,61),(71,73)
✓- Product of numbers is odd ONLY IF every factor is odd
✓- Sum of two odd numbers is always even
✓- Number of divisors of n = p^a × q^b is (a+1)(b+1)

🧠 Memory hook

"Perfect: Six and Twenty-Eight" — 6 = 1+2+3; 28 = 1+2+4+7+14. For twin primes: think of prime siblings sitting next to each other on the number line with one seat gap.

🔁 Recap

✓- Full prime list below 100 has exactly 25 numbers; memorise by decade groups
✓- Primality test: divide by primes ≤ √n; if none divide, it is prime
✓- Perfect number = sum of proper divisors; 6 and 28 are the exam-relevant ones
✓- Twin primes are prime pairs with difference 2
✓- Product is even if any factor is even; product is odd only if all factors are odd
✓- Number of divisors uses the prime factorisation exponent formula (a+1)(b+1)…

Solved Example: Identifying Number Types

The fastest route to a correct answer in SSC CGL is not just knowing the definition of a prime number — it's being able to identify one (or rule it out) in under 30 seconds without a calculator.

The Core Technique: Square Root Boundary Test

Definition: A number n is prime if and only if it has no prime factor less than or equal to √n.

This works because:

If n = a × b and both a > √n and b > √n, then a × b > n — contradiction.
So at least one factor must be ≤ √n.
Therefore, if no prime ≤ √n divides n, then n has no factor at all (other than 1 and n), so it is prime.

Practical rule:

For numbers up to 100: check divisibility by 2, 3, 5, 7 (since √100 = 10, and 7 is the largest prime below 10)
For numbers up to 200: also check 11 and 13 (since √200 ≈ 14.1)
For numbers up to 400: also check 17 and 19 (since √400 = 20)

Divisibility Quick-Checks

Before doing division, use divisibility rules to eliminate candidates instantly:

Divisor	Rule	Example
2	Last digit even	84: last digit 4 → divisible
3	Sum of digits divisible by 3	93: 9+3=12 → divisible
5	Last digit 0 or 5	95: last digit 5 → divisible
7	No simple rule; just divide
11	(Sum of odd-position digits) − (Sum of even-position digits) = 0 or 11	121: (1+1)−2=0 → divisible
13	No simple rule; just divide

Solved Example 1 (from the lesson stub)

Question: Which of the following is a prime number — 91, 97, 87, 93?

Solution:

Step 1: Examine 91. √91 ≈ 9.5 → check primes 2, 3, 5, 7.

Is 91 even? No.
Digit sum = 9+1 = 10 → not divisible by 3.
Last digit not 0 or 5 → not divisible by 5.
91 ÷ 7 = 13. Yes! So 91 = 7 × 13. Composite.

Step 2: Examine 87. √87 ≈ 9.3 → check 2, 3, 5, 7.

Digit sum = 8+7 = 15 → divisible by 3. 87 = 3 × 29. Composite.

Step 3: Examine 93. √93 ≈ 9.6 → check 2, 3, 5, 7.

Digit sum = 9+3 = 12 → divisible by 3. 93 = 3 × 31. Composite.

Step 4: Examine 97. √97 ≈ 9.8 → check 2, 3, 5, 7.

Not even. Digit sum = 9+7 = 16 → not divisible by 3. Last digit ≠ 0,5. 97 ÷ 7 = 13.857… Not divisible.
No prime ≤ √97 divides 97. 97 is prime.

Conclusion: 97 is the prime.

Why 91 is the classic trap: 91 looks prime because it is not divisible by 2, 3, or 5. But 7 × 13 = 91 — always check 7 for two-digit numbers!

Solved Example 2: Identifying the Type of a Given Number

Question: Classify: 1, 2, 4, 15, 17, 28, 36.

Solution:

Number	Factors	Classification
1	1 only	Neither prime nor composite
2	1, 2	Prime (only even prime)
4	1, 2, 4	Composite
15	1, 3, 5, 15	Composite
17	1, 17	Prime
28	1, 2, 4, 7, 14, 28	Composite; also a perfect number
36	1,2,3,4,6,9,12,18,36	Composite; perfect square (6²)

Solved Example 3: How Many Primes Between Two Numbers?

Question: How many prime numbers lie between 80 and 100?

Solution:

Step 1: Candidates: 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99.

Step 2: Eliminate composites quickly —

Even numbers: 82, 84, 86, 88, 90, 92, 94, 96, 98 → all composite
Divisible by 5: 85 (=5×17), 90, 95 (=5×19) → composite
Divisible by 3: 81 (=3⁴), 87 (=3×29), 93 (=3×31), 99 (=9×11) → composite
91 = 7 × 13 → composite

Step 3: Remaining: 83 and 89 and 97.

83: √83 ≈ 9.1. Not divisible by 2, 3, 5, 7. → Prime
89: √89 ≈ 9.4. Not divisible by 2, 3, 5, 7. → Prime
97: already confirmed prime above.

Conclusion: 3 primes between 80 and 100 (83, 89, 97).

Common SSC Traps

Trap 1 — 1 is prime: FALSE. 1 is neither prime nor composite.

Trap 2 — 91 is prime: FALSE. 91 = 7 × 13.

Trap 3 — 2 is not prime (it's even): FALSE. 2 is the only even prime.

Trap 4 — All odd numbers are prime: FALSE. 9 = 3×3, 15 = 3×5, 21 = 3×7 are all odd composites.

Trap 5 — √n must be an integer to be relevant: FALSE. Even if √n is not an integer (e.g., √91 ≈ 9.5), you still check all primes up to the floor of √n.

Worked Shortcut: Composite Identification by Digit Sum

Question: Without dividing, determine whether 561 is prime.

Solution:

Step 1: Digit sum = 5+6+1 = 12. Divisible by 3 → 561 is divisible by 3. Composite. (561 = 3 × 187)

This digit-sum trick eliminates most large numbers before you even attempt division.

🔑 Key points

✓- To test primality of n, check only primes ≤ √n — once you pass √n, stop
✓- For numbers 1–100: checking 2, 3, 5, 7 is sufficient (√100 = 10)
✓- 91 = 7 × 13 is the most common prime-impersonator in SSC exams
✓- 1 is neither prime nor composite — a guaranteed trick question
✓- Use divisibility rules (digit sum for 3; last digit for 2, 5) before attempting division
✓- Between 80 and 100 there are exactly 3 primes: 83, 89, 97
✓- Perfect numbers (6, 28) and twin primes are separate special categories

🧠 Memory hook

"Seven Traps Ninety-One" — whenever you see 91, immediately check ÷7. For the test: "Check up to the Square Root, not a step more."

🔁 Recap

✓- Primality test: find √n, then test divisibility by each prime up to that bound
✓- Apply divisibility rules first (2, 3, 5) to quickly eliminate most candidates
✓- 91 = 7×13, 87 = 3×29, 93 = 3×31 — all favourite composite traps in MCQ options
✓- 97 is the largest prime below 100
✓- 1 is not prime; 2 is prime and is the only even prime
✓- Between any two consecutive perfect squares, count primes carefully using the method above

⚡ Speed Tricks & Shortcuts

💡 Speed tricks & shortcuts

To test if n is prime, only trial-divide by primes ≤ √n (e.g. for 149, check up to 12).
2 is the only even prime; 1 is neither prime nor composite.
Two numbers are co-prime if their HCF = 1 (they needn't be prime themselves, e.g. 8 & 9).
Product of n consecutive integers is always divisible by n!.

⚠️ Common mistake

Don't call 1 a prime and don't forget 2 — "smallest prime is 2, smallest odd prime is 3" is a classic trap.

Types of Numbers — Revision Notes

Quick-revision notes for Types of Numbers — the must-know points for SSC CGL Tier-I/II.

Natural numbers (1,2,3…), Whole numbers (0,1,2…), Integers (…−2,−1,0,1,2…), and their subsets.
Rational numbers can be written as p/q (q≠0); irrational numbers (like √2, π) cannot.
Prime numbers have exactly two factors (1 and itself); 2 is the only even prime; 1 is neither prime nor composite.
Composite numbers have more than two factors; co-prime numbers share HCF 1.
Even numbers are divisible by 2; odd are not. Perfect numbers equal the sum of their proper divisors (e.g. 6 = 1+2+3).
Real numbers = rationals + irrationals; they cover the entire number line.

Types of Numbers — Flashcards (SSC CGL)

Cover the answer, recall, then check. 7 cards on the must-know Types of Numbers facts for SSC CGL.

Q1. What is the only even prime number?
A1. 2.

Q2. Is 1 a prime number?
A2. No — 1 is neither prime nor composite.

Q3. Define a rational number.
A3. A number expressible as p/q where p and q are integers and q ≠ 0.

Q4. Give an example of an irrational number.
A4. √2 (or π) — it cannot be written as a ratio of two integers.

Q5. What are co-prime numbers?
A5. Two numbers whose HCF is 1 (e.g. 8 and 9).

Q6. How many factors does a prime number have?
A6. Exactly two — 1 and the number itself.

Q7. Why is 6 called a perfect number?
A7. Because the sum of its proper divisors (1+2+3) equals 6.

Types of Numbers — Formula Sheet

Key formulas

Classification: natural ⊂ whole ⊂ integer ⊂ rational ⊂ real; irrationals are non-terminating non-repeating.
Prime: exactly two factors; composite: more than two; 1 is neither.
Co-prime: HCF = 1. Perfect number = sum of proper divisors (e.g. 6, 28).
Sum of first n naturals n(n+1)/2; first n odd numbers = n²; first n even = n(n+1).
Rational number lies between any two reals; between a and b: (a+b)/2.
A number is rational iff its decimal terminates or repeats.

🔑 Key points

✓- Prime has exactly 2 factors; 1 is neither prime nor composite.
✓- Co-prime ⇒ HCF = 1.
✓- Sum of first n odd numbers = n².
✓- Rational ⇔ terminating or repeating decimal.

Usage: identify the number type first — it fixes which divisibility/sum rules apply.

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