Let x be a positive integer such that 7x + 96 is divisible by x. How many values of x are possible?

Answer: (c) 12

7x is always a multiple of x, so x divides 7x + 96 exactly when x divides 96. Since 96 = 2⁵ × 3, it has (5 + 1)(1 + 1) = 12 divisors, so x can take 12 values.

1. 7x is divisible by x, so x divides 7x + 96 only when x divides 96.
2. 96 = 2⁵ × 3.
3. Number of divisors = (5 + 1) × (1 + 1) = 12: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96.

Remember · If x divides kx + N, then x divides N; the number of divisors of 2^a × 3^b is (a + 1)(b + 1).

Question and answer: UPSC's official GS Paper II (2023, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error