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Every calculation in the exam ultimately comes back to number properties and the order of operations. Get this chapter solid and simplification questions become free marks.
Brackets → Orders (powers/roots) → Division → Multiplication → Addition → Subtraction. Division and multiplication have equal priority — solve left to right; same for addition and subtraction.
| Divisible by | Rule |
|---|---|
| 2 | Last digit is even |
| 3 | Sum of digits divisible by 3 |
| 4 | Last 2 digits divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 9 | Sum of digits divisible by 9 |
| 11 | (Sum of digits at odd places) − (sum at even places) divisible by 11 (or 0) |
HCF (Highest Common Factor) is the largest number dividing all given numbers exactly. LCM (Lowest Common Multiple) is the smallest number divisible by all given numbers.
Key relation: for two numbers a and b, HCF × LCM = a × b.
Find the HCF and LCM of 12 and 18.
12 = 2² × 3, 18 = 2 × 3². HCF = 2 × 3 = 6. LCM = 2² × 3² = 36. Check: 6 × 36 = 216 = 12 × 18 ✓
Simplify: 15 + 6 ÷ 2 × 3 − 4
Division/multiplication first, left to right: 6 ÷ 2 = 3, then 3 × 3 = 9. Now: 15 + 9 − 4 = 20.
UdaanPath posts new SSC, Railway and Banking notifications every 6, 8 and 12 days respectively. If all three were posted today, after how many days will all three land on the same day again?
This is LCM(6, 8, 12). 6 = 2×3, 8 = 2³, 12 = 2²×3 → LCM = 2³×3 = 24 days.
Solving addition/subtraction before division/multiplication is the single biggest reason students lose marks here — always resolve ÷ and × first, strictly left to right.
1. Simplify: 20 − 4 × 2 + 6 ÷ 2
2. What is the HCF of 24 and 36?
3. UdaanPath posts SSC and Railway notifications every 6 and 8 days respectively. If both were posted today, after how many days will they next coincide?
4. Which number is divisible by 11?