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

(UPSC CSAT – 2023)

1. 6
2. 7
3. 8
4. 9

Solution: Ans.(4)

All the factors of 1155 (1186-31=1155) greater than 31 are the required numbers

How to calculate number of factors of 1155:

1155 = 11¹×7¹×5¹×3¹

Number of factors = 2×2×2×2 = 16

Factors less than 31 are = 1, 3, 5, 7, 11, 15, 21 (7 factors)

So, the remaining factors greater than 31 are = 16 – 7 = 9

Hence, the number of natural number which when divides 1186 gives 31 as remainder are 9.