CAPF AC 2020 Q120: General Mental Ability – Quantitative Aptitude
- (a) 3
- (b) 2
- (c) 1
- (d) 0
Answer: (a) 3
Explanation
Same remainder with two divisors
Given: A is the smallest positive integer which when divided by 9 and 12 leaves remainder 8.
B is the smallest positive integer which when divided by 9 and 12 leaves remainder 5.
We need the value of A − B.
How to think: A and B are built the same way.
Only the remainders differ.
So the difference is just the difference of the remainders: 8 − 5.
| Step | Working |
|---|---|
| LCM of 9 and 12 | 36 |
| Numbers leaving remainder 8 | 8, 44, 80, ... (36k + 8) |
| Numbers leaving remainder 5 | 5, 41, 77, ... (36k + 5) |
Two ways to read smallest:
| Reading | A | B | A − B |
|---|---|---|---|
| Number itself may be the remainder | 8 | 5 | 3 |
| Number bigger than the divisors | 44 | 41 | 3 |
Either way the answer is the same.
Answer: (a) 3.
Why each option is right or wrong
- (a) CORRECT. A − B = (36k + 8) − (36k + 5) = 3.
- (b) Incorrect. 2 is not the gap between the remainders 8 and 5.
- (c) Incorrect. 1 has no basis in the working.
- (d) Incorrect. 0 would mean A and B are equal, but their remainders differ.
Key point
Number leaving remainder r when divided by a and b = LCM(a, b) × k + r. LCM(9, 12) = 36. A = 36k + 8, B = 36k + 5, so A − B = 3.
Remember
Same remainder r for several divisors: number = LCM × k + r.
Common mistakes
- Taking the LCM of 9 and 12 as 108, their product; it is 36.
- Spending time finding A and B when only the difference is asked.
- Adding the remainders instead of subtracting.
Exam tip
Educated guess: the same divisors and the same rule give numbers that differ only by the remainders. 8 − 5 = 3.
Sources
- NCERT Class VI Mathematics — Playing with Numbers (LCM)
- NCERT Class X Mathematics — Real Numbers