Number System
Number system questions reward pattern recognition rather than calculation. Unit digits repeat in cycles of at most four, divisibility has fixed rules, and HCF and LCM are tied together by one identity worth memorising.
Updated 19 September 2026 · HCF/LCM and unit-digit questions are near-guaranteed.
How to think about number system
HCF and LCM are linked
For any two numbers, HCF × LCM = the product of the numbers. Given any three of those four quantities you can find the fourth immediately — which is exactly how most questions on this are framed.
Unit digits cycle with period 4 or less
The last digit of a power repeats. 2 cycles 2, 4, 8, 6; 3 cycles 3, 9, 7, 1; 7 cycles 7, 9, 3, 1. Divide the exponent by 4 and use the remainder to pick the position — remainder 0 means the last item of the cycle. Digits 0, 1, 5 and 6 never change.
Divisibility rules worth knowing cold
By 3 or 9: sum of digits. By 4: last two digits. By 8: last three digits. By 11: alternating digit sum. By 6: divisible by both 2 and 3. These turn a long division into a glance.
Formulas and shortcuts
- HCF-LCM identity
HCF × LCM = Product of the two numbers - Unit digit of a power
Find exponent mod 4, then read the position in the digit cycleA remainder of 0 means the 4th (last) item in the cycle.
- Sum of first n natural numbers
n(n + 1) / 2 - Sum of first n squares
n(n + 1)(2n + 1) / 6 - Number of factors
If N = aᵖ × bᑫ then N has (p + 1)(q + 1) factors
Solved examples
Q1. Verify the HCF-LCM identity for 12 and 18.
- 12 = 2² × 3 and 18 = 2 × 3²
- HCF takes the lowest power of each common prime: 2 × 3 = 6
- LCM takes the highest power of each prime: 2² × 3² = 36
- HCF × LCM = 6 × 36 = 216, and 12 × 18 = 216
Answer: The identity holds: both sides equal 216
Q2. Find the unit digit of 7¹⁰⁰.
- The unit digits of powers of 7 cycle: 7, 9, 3, 1 — a cycle of length 4.
- Divide the exponent by 4: 100 ÷ 4 = 25 remainder 0
- A remainder of 0 points at the last item in the cycle.
Answer: 1
Check the pattern: 7¹=7, 7²=49, 7³=343, 7⁴=2401. The 4th ends in 1, as does every 4th after it.
Q3. Find the largest three-digit number divisible by 7.
- The largest three-digit number is 999.
- 999 ÷ 7 = 142 remainder 5
- Subtract the remainder: 999 − 5 = 994
Answer: 994
994 ÷ 7 = 142 exactly.
Practice questions with answers
1. The LCM of 12 and 15 is:
Answer: C. 60 — 12 = 2²×3, 15 = 3×5. LCM = 2²×3×5 = 60.
2. The HCF of 24 and 36 is:
Answer: C. 12 — 24 = 2³×3, 36 = 2²×3². Lowest powers: 2²×3 = 12.
3. The unit digit of 2⁵⁰ is:
Answer: B. 4 — Cycle 2,4,8,6. 50 ÷ 4 leaves remainder 2 → the 2nd item → 4.
4. The smallest number exactly divisible by 6, 8 and 12 is:
Answer: B. 24 — That is the LCM: 2³×3 = 24.
5. How many factors does 36 have?
Answer: C. 9 — 36 = 2² × 3². Factors = (2+1)(2+1) = 9.
6. The sum of the first 20 natural numbers is:
Answer: C. 210 — n(n+1)/2 = 20 × 21 / 2 = 210.
7. If the HCF of two numbers is 6 and their LCM is 36, and one number is 12, the other is:
Answer: B. 18 — HCF × LCM = product → 6 × 36 = 12 × x → 216 = 12x → x = 18.
Where marks get lost
- Reading a remainder of 0 as the first item in a unit-digit cycle instead of the last.
- Swapping the rules: HCF takes the lowest powers, LCM the highest.
- Applying the HCF × LCM identity to three numbers — it only holds for two.
- Long-dividing when a divisibility rule would answer it instantly.
Reading is not practising.
Run a timed set on number system and see which step you actually lose time on. Free, no card needed.
Start a timed set