# Q. 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

(d) Infinitely many

Correct Answer: (c) 12

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

## 7x + 96 is divisible by x

The statement “7x + 96 is divisible by x” implies that when you divide the expression “7x + 96” by “x”, you get a whole number (an integer) as a result. This means that “x” must be a factor of the constant term “96”, because the “7x” term will always be divisible by “x” (since it contains “x” as a factor).

To find all the possible values of “x”, we need to list all the factors of 96. The prime factorization of 96 is indeed (2^5 * 3). From this prime factorization, we can determine all the factors of 96 by taking various combinations of these prime factors.

Here are all the factors of 96:

1,2,4,8,16,32,3,6,12,24,48,96

These are the 12 factors of 96, which means that “x” can be any of these 12 values. If “x” is any of these values, then “7x + 96” will be divisible by “x”. Therefore, the statement is correct, and there are indeed 12 possible values for “x”.