Question

The number 1953125 is equal to 5^9.
Explain how the information above informs you that 1953125 is not a square number

Answer 1

this is not a square number because it is equal to 5^9 which is not square number

Answer 2

The number 1953125 is equal to 5^9.

The information above informs that 1953125 is not a square number, and is given as follows.

Method I

If a number is a perfect square, then the prime factorization of that number should be of  $\prod_{i=1}^np_i^{2n_i}$ where pi is the ith prime number and ni ∈ N.

But we have, 1953125=5⁹, so it cannot be a perfect square.

Method II

If the last digit of a given number is 5, then the last three digits must be perfect squares, 025 or 225 or 625. Otherwise, that number cannot be a perfect square. And as 125 is not a perfect square, so no number ending with 125 can be a perfect square.