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

UPSC CSAT 2023: A Deep Dive into a Divisibility Challenge

The UPSC Civil Services Aptitude Test (CSAT) is renowned for its rigorous examination process. The 2023 CSAT featured a multitude of challenging questions, one of which was a divisibility puzzle that has puzzled many aspirants. In this blog, we will explore a thought-provoking question from the CSAT 2023 and provide a comprehensive solution.

The Question:

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

The Solution:

To tackle this problem, we need to understand what it means for “7x + 96 to be divisible by x.” For any integer a to be divisible by integer b, it should leave a remainder of 0 when divided by b. In other words, a % b should be equal to 0. [ a%b means remainder when a is divided by b]

In this context, 7x + 96 should be divisible by x, which means:

(7x + 96) % x = 0

Let’s break this down step by step:

(1) (7x % x) + (96 % x) = 0

(2) 7x % x = 0 (because 7x is a multiple of x)

(3) So, (0) + (96 % x) = 0

(4) Therefore, 96 % x = 0

This tells us that 96 must be divisible by x, which means x is a factor of 96. Now, let’s find out how many positive and negative integer factors 96 has.

Factors of 96:

To find the factors of 96, we list all the pairs of numbers, both positive and negative, that multiply to give 96. These factors are:

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

So, there are a total of twelve factors of 96. These numbers represent the possible values of x that satisfy the given condition.

Answer:

There are twelve possible values for x: which are factors of 96.

In conclusion, the CSAT 2023 question about the divisibility of 7x + 96 by x might appear complex at first, but by breaking it down step by step, we can find a straightforward solution. Understanding the concept of divisibility and recognizing that x must be a factor of 96 allows us to identify the possible values for x. This problem serves as an example of how critical thinking and mathematical reasoning are essential skills for tackling UPSC CSAT questions.

If you have any more questions or need further clarification, please feel free to ask!

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