# Q. How many natural numbers are there which give a remainder of 31 when 1186 is divided by these natural numbers?

(a) 6

(b) 7

(c) 8

(d) 9

Correct Answer: (d) 9

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

The remainder when a number is divided by another number is the difference between the dividend and the largest multiple of the divisor that is less than or equal to the dividend.

In this case, we are given that the

## remainder is 31 when 1186 is divided by a certain number.

This means that the number we are looking for must be a divisor of (1186 – 31) = 1155.

The divisors of 1155 are 1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155.

However, since the remainder is 31, the divisor must be greater than 31.

So, the possible numbers are **33, 35, 55, 77, 105, 165, 231, 385, 1155.**

Hence, there are 9 such natural numbers. So, the correct answer is (d) 9.