The difference between any two natural numbers is 10. What can be said about the natural numbers which are divisible by 5 and lie between these two numbers ?

(a) There is only one such number.
(b) There are only two such numbers.
(c) There can be more than one such number. ✓ UPSC's answer
(d) No such number exists.

Why the answer is (c)

• Let the two natural numbers be $x$ and $x+10$, where $x$ is a natural number.

• The numbers strictly between them are $x+1, x+2, \dots, x+9$.

• We need to find how many of these 9 numbers are divisible by 5.

• If $x$ is a multiple of 5 (e.g., $x=5$), the numbers between are $6, 7, 8, 9, 10, 11, 12, 13, 14$. Only 10 is divisible by 5 (count = 1).

• If $x$ is not a multiple of 5 (e.g., $x=1$), the numbers between are $2, 3, 4, 5, 6, 7, 8, 9, 10$. Both 5 and 10 are divisible by 5 (count = 2).

• Since the count can be either 1 or 2 depending on the starting number, it is not fixed to a single value, meaning there can be more than one such number in some cases (specifically, two).

Why the other options are wrong

(a) There is only one such number.: This is incorrect because if the smaller number is not a multiple of 5 (e.g., 1 and 11), there are two numbers (5 and 10) divisible by 5 between them.
(b) There are only two such numbers.: This is incorrect because if the smaller number is a multiple of 5 (e.g., 5 and 15), there is only one number (10) divisible by 5 between them.
(d) No such number exists.: This is incorrect because there is always at least one multiple of 5 between any two numbers differing by 10 (e.g., 10 is between 5 and 15).

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.