Introduction
Number System covers the properties of integers — divisibility, factors, multiples, LCM, HCF, and remainders.
It rarely appears as a standalone heavy topic in banking prelims but its rules are used constantly inside
Simplification, DI, and word-problem questions.
Divisibility Rules
| Divisible by | Rule |
|---|
| 2 | Last digit is even |
| 3 | Sum of digits divisible by 3 |
| 4 | Last two digits divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 8 | Last three digits divisible by 8 |
| 9 | Sum of digits divisible by 9 |
| 11 | Difference of (sum of digits at odd places) and (sum at even places) is 0 or divisible by 11 |
LCM and HCF Shortcuts
- Product Rule: LCM × HCF = Product of the two numbers (for exactly two numbers).
- HCF of numbers = largest number that divides all of them exactly.
- LCM of numbers = smallest number divisible by all of them.
- For fractions: LCM (of fractions) = LCM(numerators) / HCF(denominators); HCF (of fractions) = HCF(numerators) / LCM(denominators).
Q. Find the HCF and LCM of 36 and 60.
36 = 2²×3², 60 = 2²×3×5. HCF = common lowest powers = 2²×3 = 12.
LCM = highest powers of all primes = 2²×3²×5 = 180. Check: 12×180 = 2160 = 36×60 ✓.
Q. What is the remainder when 2⁵⁰ is divided by 7?
Find the cycle of remainders of powers of 2 mod 7: 2¹=2, 2²=4, 2³=1, then it repeats every 3 steps.
50 ÷ 3 leaves remainder 2, so 2⁵⁰ behaves like 2² mod 7 → remainder = 4.
⚠️ Key Insight: For remainder problems, always look for a repeating cycle in the powers rather
than calculating the full large number — cycles for powers of 2 through 9 rarely exceed length 6.
💡 Exam Tip: Memorise squares up to 30² and cubes up to 15² — this single habit speeds up
Number System, Simplification, and Quadratic Equations together.
✅ Practice Focus: Divisibility rules for 3, 9, 11 · LCM-HCF word problems · Remainder cycles for
powers.