prove ta number which has 101 after the units place cannot be a square number?

Question

i.e .........................101x, x is any natural number (from 0 to 9)cannot be a square number

Answer 1

here's a proof:

first, note that the square of every natural number mod 4 is either 0 or 1, i.e., the square of every natural number is either a multiple of 4, or a multiple of 4 plus 1. so, if .....101x is a square, then either 1x should be a multiple of 4, or a multiple of 4 plus 1. therefore, x is one of these numbers:

2, 3, 6, 7.

Now, similarly observe that the square of a natural number mod 5 is either 0, 1, or 4 (this is because in mod 5, 0^2=0, 1^2=1, 2^2=4, 3^2=4, 4^2=1). Therefore in order for ....101x to be a square, we must have 1x = 0 or 1 or 4 mod 5. The only value of x out of the above x that satisfies this is

x = 6.

Now, we know that this number ends in 1016. However, this means that the number is divisible by 8, but not divisible by 16. Therefore, it cannot be a perfect square, since if N^2 is divisible by 8, it means that N is divisible by 4 and therefore N^2 must also be divisible by 16.

Answer 2

Because there is no square between 961 and 1024

Sorry I realise I didn't understand the question. The next answer is perfect.