CAPF AC 2025 Q67: General Mental Ability – Quantitative Aptitude
The sum of two numbers is 168. If their HCF is 14, how many such pairs of numbers are possible?
- (a) 1
- (b) 2
- (c) 3
- (d) 0
Answer: (b) 2
Explanation
Pairs with sum 168 and HCF 14
Given: two numbers add to 168 and their HCF is 14.
How to think: write the numbers as 14x and 14y, where x and y share no common factor (they are co-prime).
Then 14x + 14y = 168, so x + y = 12.
| x | y | Co-prime? | Pair of numbers |
|---|---|---|---|
| 1 | 11 | Yes | 14 and 154 |
| 2 | 10 | No (both even) | not allowed |
| 3 | 9 | No (both divisible by 3) | not allowed |
| 4 | 8 | No | not allowed |
| 5 | 7 | Yes | 70 and 98 |
| 6 | 6 | No (equal) | not allowed |
Two pairs are possible: (14, 154) and (70, 98).
Answer: (b) 2.
Why each option is right or wrong
- (a) Incorrect. Two pairs work, not one.
- (b) CORRECT. The pairs (14, 154) and (70, 98) are the only two.
- (c) Incorrect. Only (1, 11) and (5, 7) are co-prime among the splits of 12.
- (d) Incorrect. Pairs do exist, such as (14, 154).
Key point
HCF h: numbers hx, hy with x, y co-prime. Here x + y = 168 ÷ 14 = 12; co-prime pairs (1, 11) and (5, 7) give the pairs (14, 154) and (70, 98).
Remember
168 ÷ 14 = 12: co-prime splits 1 + 11 and 5 + 7 give 2 pairs.
Common mistakes
- Counting pairs like (28, 140) whose HCF is 28, not 14.
- Counting (x, y) and (y, x) as different pairs.
- Forgetting to divide 168 by the HCF first.
Exam tip
Educated guess: 12 splits into 1 + 11, 5 + 7 and the even or multiples-of-3 splits. Only splits where the two parts share no factor work, and only two of the five splits are like that.
Sources
- NCERT Class X Mathematics, Chapter 1 (Real Numbers: HCF and LCM)
- NCERT Class VI Mathematics, Chapter 3 (Playing with Numbers: common factors)