250 has the prime factorization 2 * 5 * 5. The greatest common divisor of 250 and 1000 is 20, so the smallest possible value of n is 20.

The largest perfect square that’s less than or equal to 1000 is 1001 (1001 = 10^2 + 1 = 100 + 1). So we want to find a number between 1 and 100 that’s a multiple of 20, but not divisible by 10.

We can do this by looking at the prime factorization of 1001:

1001 = 2^5 * 5^2 * 7^2 * 11^2

This means that 1001 can be written as an integer times either 2^5 times 5^2 or 2^5 times 7^2 (and possibly some other combinations). So what we need to do is find all of the factors of 1001 that are also factors of 20. There are two: 2 and 7 (and possibly 11).

This means that 1001 can be written as an integer times either 2 times 5 or 7 times 2 (and possibly some other combinations).

Since we want a number between 1 and 100 that’s a multiple of 20, but not divisible by 10 or any combination thereof.

Last modified: October 2, 2022