-  On dividing 13294 by 97, we get remainder = 5.
Required number to be subtracted = 5

To find the least number that must be subtracted from 13294 so that the remainder is exactly divisible by 97, we need to use the concept of modular arithmetic.

Modular arithmetic is a branch of mathematics that deals with operations on remainders. It is also known as clock arithmetic since it is used to solve problems related to time and periodic phenomena.

Let x be the number that we need to subtract from 13294 to get a remainder that is exactly divisible by 97. Then we can write

13294 - x ≡ 0 (mod 97)

This means that the remainder when 13294 is divided by 97 is the same as the remainder when x is divided by 97. In other words, we are looking for a number x that leaves a remainder of 0 when divided by 97.

To solve for x, we can use the fact that if a ≡ b (mod m), then a - b is divisible by m. Therefore, we have:

13294 - x ≡ 0 (mod 97)

⇒ 13294 ≡ x (mod 97)

⇒ x = 13294 - 97k for some integer k

We want to find the smallest positive value of k that makes x divisible by 97. This means that 13294 - 97k should be the smallest positive integer that is greater than or equal to 0 and is divisible by 97. We can find this value as follows:

13294 - 97k ≥ 0

⇒ k ≤ 136.64

Since k must be an integer, the largest integer that is less than or equal to 136.64 is 136. Therefore, we can take k = 136 and compute:

x = 13294 - 97k

= 13294 - 97(136)

= 13294 - 13192

= 102

Therefore, the least number that must be subtracted from 13294 so that the remainder is exactly divisible by 97 is 102. However, the question asks for the least positive number, which is 5. This is because if we subtract 97 from 102, we get 5, which is the smallest positive number that gives a remainder that is exactly divisible by 97.

Therefore, the correct answer is option D, 5.

If you think the solution is wrong then please provide your own solution below in the comments section .