Number System — Revision Notes
Number System underpins all of AFCAT Quant: divisibility, HCF/LCM, factors and remainders. These give fast marks and also speed up other topics. The rules are fixed, so a short set of divisibility tests and HCF/LCM methods pays off across the whole section.
Divisibility rules (high-yield)
- 2: last digit even. 3: digit sum divisible by 3. 4: last two digits divisible by 4.
- 5: ends in 0 or 5. 6: divisible by 2 and 3. 8: last three digits divisible by 8.
- 9: digit sum divisible by 9. 11: difference of alternate-digit sums is 0 or a multiple of 11.
HCF and LCM
- HCF (Highest Common Factor) = largest number dividing all; LCM (Lowest Common Multiple) = smallest number all divide into.
- Key identity: HCF x LCM = product of the two numbers. For 12 and 18: HCF = 6, LCM = 36, and 6 x 36 = 216 = 12 x 18.
- Prime factorisation: 12 = 2^2 x 3, 18 = 2 x 3^2; HCF = 2 x 3 = 6, LCM = 2^2 x 3^2 = 36.
Types and facts
- Primes have exactly two factors (2 is the only even prime). Natural, whole, integer, rational, irrational numbers form the hierarchy.
- Sum of first n natural numbers = n(n+1)/2; sum of first n odd numbers = n^2.
Exam Tricks & Tips
- 🎯 Memorise divisibility rules - they cut brute-force division on large numbers.
- 🎯 Use HCF x LCM = product of the two numbers to find a missing value instantly.
- 🎯 For LCM/HCF, prime-factorise: HCF takes the lowest powers, LCM the highest.
- 🎯 Sum of first n naturals = n(n+1)/2 - e.g. 1 to 100 = 100 x 101 / 2 = 5050.
- 🎯 2 is the only even prime; remember it for "how many primes" traps.
- ❌ Common mistake: swapping HCF and LCM - HCF is the largest common divisor (small), LCM is the smallest common multiple (large).
Quick recap
Know the divisibility rules, the HCF x LCM = product identity, prime factorisation for HCF/LCM, and the natural-number sum formulas. Number System is rule-based and fast - a reliable Quant scorer.
Number System — Flashcards
Cover the answer, recall, then check. 12 number-system cards for AFCAT.
Q1. Divisibility rule for 3?
A1. The digit sum is divisible by 3.
Q2. Divisibility rule for 9?
A2. The digit sum is divisible by 9.
Q3. Divisibility rule for 11?
A3. The difference of alternate-digit sums is 0 or a multiple of 11.
Q4. State HCF x LCM in terms of two numbers.
A4. HCF x LCM = product of the two numbers.
Q5. HCF and LCM of 12 and 18?
A5. HCF = 6, LCM = 36 (and 6 x 36 = 216 = 12 x 18).
Q6. In prime factorisation, HCF takes which powers?
A6. The lowest powers of the common primes (LCM takes the highest).
Q7. Which is the only even prime number?
A7. 2.
Q8. Sum of the first n natural numbers?
A8. n(n+1)/2.
Q9. Sum of 1 to 100?
A9. 100 x 101 / 2 = 5050.
Q10. Sum of the first n odd numbers?
A10. n^2.
Q11. Divisibility rule for 4?
A11. The last two digits form a number divisible by 4.
Q12. HCF vs LCM — which is the larger value usually?
A12. The LCM (smallest common multiple, but larger number); HCF is the largest common divisor (smaller number).
Number System
The number system underpins the whole Numerical section — divisibility, remainders, HCF/LCM and place value all live here. AFCAT keeps it at a fundamentals level, so a handful of rules and a clear grasp of factors will carry you through both direct questions and the hidden number-system steps inside other problems.
Core idea / what this tests: understanding of number types, divisibility rules, factors, HCF and LCM, and remainder logic — the toolkit behind fast arithmetic.
Deep explanation
Beginner — number types and divisibility
- Types: natural, whole, integers, rational (p/q), irrational (√2, π), prime (exactly two factors), composite. 1 is neither prime nor composite; 2 is the only even prime.
- Divisibility rules: by 2 (even), 3 (digit sum ÷ 3), 4 (last two digits ÷ 4), 5 (ends 0/5), 6 (÷2 and ÷3), 8 (last three digits ÷ 8), 9 (digit sum ÷ 9), 11 (alternating digit sum ÷ 11).
Intermediate — HCF and LCM
| Concept | Meaning | Method |
|---|---|---|
| HCF | largest common factor | product of lowest powers of common primes |
| LCM | smallest common multiple | product of highest powers of all primes |
Key identity: HCF × LCM = product of the two numbers. So if HCF(a,b)=6 and LCM=72, then a×b = 432. Use HCF for "largest tile / greatest number that divides" problems and LCM for "bells ring together / smallest number divisible by all."
Advanced — remainders and prime factorisation
- Prime factorisation is the master tool: 360 = 2³ × 3² × 5. The number of factors = (3+1)(2+1)(1+1) = 24.
- Remainders: to find the remainder of a large product/power, take remainders step by step. Remainder of a number mod 9 = its digit sum mod 9.
- Unit-digit patterns: powers cycle (2: 2,4,8,6; 3: 3,9,7,1). To find the unit digit of 7⁴⁵, use the cycle of 7 (7,9,3,1, period 4): 45 mod 4 = 1 → unit digit 7.
Worked example
Q. The HCF of two numbers is 8 and their LCM is 96. If one number is 32, find the other.
Use HCF × LCM = product of the numbers: 8 × 96 = 32 × other. So other = (8 × 96) / 32 = 768 / 32 = 24.
Answer: 24. (Check: HCF(32,24)=8 ✓, LCM(32,24)=96 ✓.) The identity turns a wordy question into one line.
AFCAT relevance
Number-system questions appear directly in AFCAT Numerical Ability and, more importantly, as hidden steps inside ratio, time-and-work and simplification problems (finding LCMs, checking divisibility). A firm grip here speeds up the entire section.
Speed tricks and shortcuts
- HCF × LCM = product of the two numbers — solves most such questions in one step.
- Digit sum gives divisibility by 3 and 9, and the remainder mod 9.
- Unit-digit cycles (period 4 for most digits) find last digits of big powers.
- Number of factors = product of (exponent + 1) from the prime factorisation.
Swapping the HCF and LCM methods — taking highest powers for HCF or lowest powers for LCM. HCF uses the lowest powers of common primes; LCM uses the highest powers of all primes. Reversing them is a frequent slip.
- ✓- 1 is neither prime nor composite; 2 is the only even prime.
- ✓- HCF = lowest powers of common primes; LCM = highest powers of all primes.
- ✓- HCF × LCM = product of the two numbers.
- ✓- Number of factors = product of (exponents + 1).
- ✓- Unit digits of powers repeat in cycles (period 4 for most).
- ✓The number system is the section's toolkit: know your divisibility rules, use prime factorisation for factors and remainders, and lean on HCF × LCM = product to crack pair problems in one line. Solid fundamentals here accelerate everything downstream.
Number System — Formula Sheet
Key formulas
- Natural, whole, integer, rational (p/q), irrational, real, prime, composite numbers.
- Sum of first n naturals = n(n+1)/2; of squares = n(n+1)(2n+1)/6; of cubes = [n(n+1)/2]².
- Even + even = even; odd + odd = even; product of two odds is odd.
- A prime > 3 is of the form 6k ± 1.
- Number of primes is infinite; 2 is the only even prime.
- (aⁿ − bⁿ) is divisible by (a − b) for all n; by (a + b) for even n.
- ✓- Σn = n(n+1)/2, Σn² = n(n+1)(2n+1)/6, Σn³ = [n(n+1)/2]².
- ✓- Prime > 3 ⇒ 6k ± 1 form.
- ✓- aⁿ − bⁿ divisible by a − b always.
- ✓- 2 is the only even prime.
Usage: use the standard series sums and divisibility identities to shortcut number problems.
Number System — Worked Example
Worked Example
Problem: Find the unit (last) digit of 7³⁵.
Solution:
The last digit of a power repeats in a fixed cycle. Find the cycle of unit digits for powers of 7:
7¹ = 7 → 7
7² = 49 → 9
7³ = 343 → 3
7⁴ = 2401 → 1
7⁵ → 7 again …
So the unit digits cycle as 7, 9, 3, 1 and repeat every 4 powers (cycle length = 4).
To find where 7³⁵ falls in the cycle, divide the exponent by the cycle length and take the remainder:
35 ÷ 4 = 8 remainder 3.
A remainder of 3 means the unit digit is the 3rd term of the cycle (7, 9, 3, 1), which is 3.
(If the remainder had been 0, we would take the last term of the cycle, i.e. the 4th.)
Answer: The unit digit of 7³⁵ is 3.
- ✓- Unit digits of powers repeat in cycles (for 7: 7, 9, 3, 1, length 4).
- ✓- Divide the exponent by the cycle length; the remainder locates the digit.
- ✓- Remainder 0 corresponds to the last digit of the cycle.