** Any**integer number can be expressed as

`7x +a` where `a` is integer `in (0,6)`

Since * all *perfect squares are the square of particular integer numbers then

**perfect squares `P` can be expressed as**

*all*`P = (7x+a)^2`

Now, expanding this out we get

`P = 49x^2 + 14ax + a^2 = 7(7x^2 + 2ax) + a^2`

The remainder upon dividing `P` by 7 will then be `a^2` modulo 7.

Since ` a` can only take the values 0,1,2,3,4,5,6 then `a^2` can only take the values 0,1,4,9,16,25,36 and `a^2` *modulo 7 *can only take the values

0,1,4,2,2,4,1.

Therefore the remainder upon dividing a perfect square `P` by 7 can only take the values 0,1,2,4 unlike other integers that can also take the values 3,5,6.

**Answer as proved above.**