1. Number System: Divisibility, HCF, LCM, Remainders and Unit Digit
Almost every aptitude paper in SSC, Railway and Bank exams starts with number questions. If you know the divisibility rules, the HCF and LCM methods, and the unit digit trick, you can answer these questions in under 30 seconds. This lesson teaches each rule with simple worked examples.
Types of numbers and divisibility rules
- Natural numbers: 1, 2, 3 ... Whole numbers: 0, 1, 2, 3 ... Integers include negative numbers too. 1 is neither prime nor composite. 2 is the only even prime number.
- Divisible by 2: last digit is even. By 5: last digit is 0 or 5. By 10: last digit is 0.
- Divisible by 3: sum of digits is divisible by 3. By 9: sum of digits is divisible by 9. Example: 5,724 has digit sum 5+7+2+4 = 18, so it is divisible by both 3 and 9.
- Divisible by 4: the last two digits form a number divisible by 4. By 8: the last three digits form a number divisible by 8. Example: 7,136 → 136 ÷ 8 = 17, so it is divisible by 8.
- Divisible by 11: (sum of digits at odd places) − (sum of digits at even places) is 0 or a multiple of 11. Example: 61,809 → (6 + 8 + 9) − (1 + 0) = 23 − 1 = 22, so it is divisible by 11.
- Divisible by 6: divisible by both 2 and 3. By 12: divisible by both 3 and 4. Use co-prime pairs only (for 12 use 3 and 4, not 2 and 6).
HCF and LCM with worked examples
- HCF (Highest Common Factor) is the biggest number that divides all the given numbers. LCM (Lowest Common Multiple) is the smallest number that all the given numbers divide.
- Prime factor method: write each number as a product of primes. HCF = common primes with the LOWEST powers. LCM = all primes with the HIGHEST powers.
- Worked example: 36 = 2² × 3² and 48 = 2⁴ × 3. HCF = 2² × 3 = 12. LCM = 2⁴ × 3² = 16 × 9 = 144.
- For two numbers only: HCF × LCM = first number × second number. Check: 12 × 144 = 1,728 and 36 × 48 = 1,728.
- Fractions: HCF of fractions = HCF of numerators ÷ LCM of denominators. LCM of fractions = LCM of numerators ÷ HCF of denominators.
- Worked example: find the least number which, when divided by 6, 8 and 12, leaves remainder 3 in each case. Step 1: LCM of 6, 8, 12 = 24. Step 2: add the remainder: 24 + 3 = 27.
Remainders and unit digit
- Remainder of a product: find the remainder of each number first, multiply them, then take the remainder again. Example: 17 × 23 ÷ 5 → remainders 2 and 3 → 2 × 3 = 6 → 6 ÷ 5 leaves remainder 1.
- Remainder of a sum works the same way: add the separate remainders, then divide again.
- Unit digit of a power repeats in a cycle of 4. Cycles: 2 → 2, 4, 8, 6; 3 → 3, 9, 7, 1; 7 → 7, 9, 3, 1; 8 → 8, 4, 2, 6.
- Numbers ending in 0, 1, 5 or 6 always keep the same unit digit. 4 gives 4 (odd power) or 6 (even power). 9 gives 9 (odd power) or 1 (even power).
- Worked example: unit digit of 7⁴⁵. Step 1: 45 ÷ 4 leaves remainder 1. Step 2: the 1st value in the cycle of 7 is 7. Answer: 7. If the remainder is 0, take the 4th value of the cycle.
- Common trap: for unit digit, look only at the last digit of the base. 1,237⁴⁵ has the same unit digit as 7⁴⁵.
Quick facts
- Divisible by 3 / 9
- Digit sum divisible by 3 / 9
- Divisible by 4 / 8
- Last 2 digits ÷ 4 / last 3 digits ÷ 8
- Divisible by 11
- Odd-place sum − even-place sum = 0 or multiple of 11
- Two numbers
- HCF × LCM = product of the numbers
- Same remainder r
- Answer = LCM + r
- Unit digit cycle
- Divide the power by 4; remainder 0 means the 4th value
💡 In the exam, test the answer options with divisibility rules before doing long division. Often three options fail the rule in a few seconds.
📝 5-question quiz on this lesson: enrol free to take it and track your score.
