Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible ?
- (a) 10
- (b) 11
- (c) 12 ✓ UPSC's answer
- (d) Infinitely many
Why the answer is (c)
• (7x + 96) ÷ x = 7 + 96 ÷ x, so x must divide 96 exactly.
• 96 = 2⁵ × 3, so the number of positive divisors is (5 + 1) × (1 + 1) = 12.
• They are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48 and 96.
• Hence 12 values, option (c).
Why the other options are wrong
- (a) 10
- Option (a) is incorrect because it undercounts the total number of divisors of 96, which is 12, not 10.
- (b) 11
- Option (b) is incorrect because it undercounts the total number of divisors of 96, which is 12, not 11.
- (d) Infinitely many
- Option (d) is incorrect because x must be a divisor of the fixed constant 96, resulting in a finite set of solutions rather than infinitely many.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.