Classification of Numbers

Numbers are grouped step by step. Natural numbers (N): 1, 2, 3... Whole numbers (W): 0, 1, 2, 3... (just add 0 to N). Integers (Z): ...-2, -1, 0, 1, 2... (positive, negative and zero). Rational numbers (Q): any number written as p/q where q is not 0, like 3/4, 0.5, -7. Irrational numbers: cannot be written as p/q, like root2, pi, root3 (non-terminating, non-repeating decimals). Real numbers (R): all rational + irrational together. Prime: exactly two factors (1 and itself), e.g. 2, 3, 5, 7. Composite: more than two factors, e.g. 4, 6, 8, 9. Remember: 1 is NEITHER prime nor composite, and 2 is the only EVEN prime.

Even numbers are divisible by 2 (0, 2, 4, 6...); odd numbers are not (1, 3, 5, 7...). Key memory tricks: even + even = even, odd + odd = even, even + odd = odd. even x anything = even; odd x odd = odd. Zero (0) is an even number, is a whole number, and is neither positive nor negative. The smallest prime is 2; the smallest composite is 4; the smallest odd prime is 3. Co-prime numbers have HCF = 1 (e.g. 8 and 15) even if individually composite. Consecutive integers always alternate even-odd.

Counting primes in a range is one of those SSC CHSL questions that looks like it needs memorisation but actually rewards a quick divisibility test. Master the under-100 prime check once and you will never freeze on questions like "How many primes lie between 10 and 30?" or "How many primes are there between 50 and 70?"

Definition: A prime number is a natural number greater than 1 that has exactly two distinct positive divisors — 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...

Definition: A composite number is a natural number greater than 1 that has more than two positive divisors. 1 is neither prime nor composite.

The worked example

Question: How many prime numbers lie between 10 and 30?

Solution:
Step 1: List all integers from 11 to 29 (the strictly-between interpretation): 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29.
Step 2: Skip every even number — they are divisible by 2 and so cannot be prime (except 2 itself, which is not in this range). That eliminates 12, 14, 16, 18, 20, 22, 24, 26, 28.
Step 3: From the odd numbers left — 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 — eliminate multiples of 3: 15 (3×5), 21 (3×7), 27 (3×9).
Step 4: Eliminate multiples of 5: 25 (5×5).
Step 5: What remains: 11, 13, 17, 19, 23, 29. Each is checked against 7 — none is divisible by 7. So all six are prime.
Conclusion: There are 6 primes between 10 and 30.

Answer: 6.

Why this technique works — the √n rule

Why it matters: to test whether a number n is prime, you do NOT have to divide by every integer from 2 to n−1. You only have to try divisors up to √n. The reason is that if n has any divisor greater than √n, it must also have a paired divisor smaller than √n (because divisors come in pairs whose product is n). So if you find no small divisor, there is no large one either.

For n < 100, √n < 10, so the only primes you must check as potential divisors are 2, 3, 5 and 7. That is it. This is the single most useful shortcut in SSC prime-counting questions.

For n < 121 (since 11² = 121), still 2, 3, 5, 7 suffice. For n < 169 (13² = 169), include 11. For n < 289 (17² = 289), include 13. Memorise these tiers.

The under-100 cheat list

Real-world example: every SSC aspirant should be able to recite the 25 primes under 100 from memory:

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.

That is 25 primes — a number worth memorising. Then:

• 1–10: 4 primes (2, 3, 5, 7)
• 11–20: 4 primes (11, 13, 17, 19)
• 21–30: 2 primes (23, 29)
• 31–40: 2 primes (31, 37)
• 41–50: 3 primes (41, 43, 47)
• 51–60: 2 primes (53, 59)
• 61–70: 2 primes (61, 67)
• 71–80: 3 primes (71, 73, 79)
• 81–90: 2 primes (83, 89)
• 91–100: 1 prime (97)

Notice the pattern flips on 91 = 7 × 13. Many students wrongly call 91 prime — it is a notorious SSC trap.

Another worked example

Question: How many prime numbers lie between 50 and 70?

Solution:
Step 1: Numbers 51 through 69.
Step 2: Strip out evens — leaves 51, 53, 55, 57, 59, 61, 63, 65, 67, 69.
Step 3: Strip out multiples of 3 (digit sum divisible by 3): 51 (5+1=6), 57 (5+7=12), 63 (6+3=9), 69 (6+9=15). Removes 51, 57, 63, 69.
Step 4: Strip out multiples of 5: 55, 65. Both end in 5.
Step 5: Remaining: 53, 59, 61, 67. Check divisibility by 7: 53/7 ≈ 7.5, 59/7 ≈ 8.4, 61/7 ≈ 8.7, 67/7 ≈ 9.6 — none divides evenly.
Conclusion: Primes between 50 and 70 are 53, 59, 61, 67 — that is 4 primes.

Common misconception

Common misconception: "1 is a prime number." Wrong. By definition a prime must have exactly two distinct divisors. The number 1 has only one divisor (itself), so it is neither prime nor composite. SSC has tested this.

A second misconception: "All odd numbers greater than 1 are prime." Wrong. 9 = 3×3, 15 = 3×5, 21 = 3×7, 25 = 5×5, 27 = 3×9, 33 = 3×11, 35 = 5×7, 39 = 3×13, 49 = 7×7, 51 = 3×17, 55 = 5×11, 57 = 3×19, 63 = 9×7, 65 = 5×13, 69 = 3×23, 77 = 7×11, 85 = 5×17, 87 = 3×29, 91 = 7×13, 93 = 3×31, 95 = 5×19. All are odd composites.

A third misconception: "2 is not prime because it's even." Wrong. 2 is prime — it is the only even prime, and the smallest prime.

Quick divisibility tests for the under-100 check

To speed up prime testing, drill these one-glance tests:

• By 2 — last digit is 0, 2, 4, 6 or 8.
• By 3 — sum of digits is divisible by 3.
• By 5 — last digit is 0 or 5.
• By 7 — no simple universal rule; just divide. For two-digit numbers, multiples of 7 are 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98 — worth memorising.

If a number under 100 escapes all four tests, it is prime.

In SSC CHSL Tier 1 you face 25 Quant questions in 60 minutes (shared with 3 other sections). Number System opens almost every paper — expect 1–2 direct questions on number types, primes or perfect/co-prime numbers, and the concepts feed HCF/LCM, remainders and divisibility. These are pure-recall marks: know the definitions and you answer in seconds.

The number family

Type	Meaning	Example
Natural (N)	counting numbers	1, 2, 3…
Whole (W)	naturals + 0	0, 1, 2…
Integers (Z)	…−2,−1,0,1,2…	−3, 0, 5
Rational (Q)	p/q, q≠0	3/4, 0.25, −2
Irrational	non-terminating non-repeating	√2, π
Prime	exactly 2 factors	2, 3, 5, 7
Composite	more than 2 factors	4, 6, 8, 9
Co-prime	HCF = 1 (as a pair)	(8, 15)

Must-know facts

• 1 is neither prime nor composite. Smallest prime = 2 (the only even prime). Smallest composite = 4.
• Primes 1–100 = 25; primes 1–50 = 15. Memorise the list.
• Perfect number = sum of its proper factors: 6 (1+2+3), 28 (1+2+4+7+14).
• Every prime > 3 is of the form 6k ± 1 (handy for quick prime-checking).

Exam Tricks & Tips

• 🎯 To test if n is prime, only check divisibility by primes up to √n. For 149, test up to 12 → 2,3,5,7,11; none divide it, so 149 is prime.
• 🎯 Any prime > 3 gives remainder 1 or 5 when divided by 6 — a fast prime filter.
• 🎯 Product of two co-primes has HCF 1; consecutive integers are always co-prime.
• 🎯 The digit-count of primes: 25 in first 100, 168 in first 1000 — occasionally asked directly.
• 🎯 Zero is a whole number and an even integer, but not natural — a classic trap option.
• ❌ Don't call 1 a prime, and don't call 0 a natural number — these two wrong assumptions cause most classification errors.

Expected exam pattern

Direct: "How many prime numbers are there between 50 and 80?", "Which of the following is a perfect number?", or identifying rational vs irrational. Single-step, ~20 seconds each.

Quick recap

Learn the seven number types, the prime list to 100 (25 primes), the 6k±1 rule, and remember 1 is neither prime nor composite while 0 is whole but not natural.

Cover the answer, recall, then check. 11 cards on number classification for SSC CHSL.

Q1. How many prime numbers are there from 1 to 100?
A1. 25 primes.

Q2. Which is the only even prime number?
A2. 2 — every other even number is divisible by 2, so composite.

Q3. Is 1 prime or composite?
A3. Neither. 1 has exactly one factor, so it is neither prime nor composite.

Q4. Define a perfect number and give the two smallest.
A4. A number equal to the sum of its proper factors: 6 (1+2+3) and 28 (1+2+4+7+14).

Q5. What does "co-prime" mean?
A5. Two numbers whose HCF = 1, e.g. (8, 15). They share no common factor except 1.

Q6. Every prime greater than 3 can be written in what form?
A6. 6k ± 1 (so it leaves remainder 1 or 5 when divided by 6).

Q7. To check whether 149 is prime, up to which number must you test divisors?
A7. Up to √149 ≈ 12, i.e. primes 2, 3, 5, 7, 11. None divide 149, so it is prime.

Q8. Is 0 a natural number?
A8. No. 0 is a whole number and an even integer, but not a natural number.

Q9. Smallest composite number?
A9. 4.

Q10. How many primes lie between 1 and 50?
A10. 15 (2,3,5,7,11,13,17,19,23,29,31,37,41,43,47).

Q11. √2 belongs to which category?
A11. Irrational — it is non-terminating and non-repeating, so it cannot be written as p/q.