UPSC Prelims 2023 CSAT Paper II · Q25 of 77 Basic Numeracy easy

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

  1. (a) 10
  2. (b) 11
  3. (c) 12 ✓ UPSC's answer
  4. (d) Infinitely many

Why the answer is (c)

• The condition that 7x + 96 is divisible by x implies that (7x + 96)/x must be an integer.

• Simplifying the expression gives 7 + 96/x, which means 96/x must be an integer.

• Therefore, x must be a positive integer divisor of 96.

• The prime factorization of 96 is 2^5 * 3^1.

• The number of divisors is calculated as (5 + 1) * (1 + 1) = 6 * 2 = 12.

• Thus, there are exactly 12 possible values for x, corresponding to 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.

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