Question

what should be multiplied to number 103875 to make it a perfect square

Answer 1

I don't know the SMALLEST number that should be multiplied to 103875 to make it a perfect square. But the most obvious answer here is to multiply it by 103875 (the number itself), since
103875 * 103875 = 103875 ^ 2

Answer 2

any square contains at least a PAIR of equal factors
e.g 1600 = 5x5 x 8x8
so ...
103875 = 25x4155
= 25 x 5 x 831
= 25 x 5 x 3 x 277 , and 277 is prime

so to make pairs, we need a 5, a 3, and a 277
5(3)(277) = 4155

so if 103875 is multiplied by 4155 the result is a perfect square, and it is the smallest multiplier

check:
103875(4155) = 431600625
and √431600625 = 20775