Question

show that the square of any positive integer cannot be the form of 5q+2 or 5q+3 for any Integer q

Answer 1

Let 'n' be any positive integer. When we divide 'n' by 5 the remainder either 0 or 1,2,3,4. So, 'n' can be written as n = 5m or n = 5m+1 or n = 5m+2 or n = 5m+3 or n = 5m+4

Those, we have the following cases:-

Answer 2

Let a^2 = 5 q + 1 for some integer q. a is a positive integer.

(a + 1) (a - 1) = 5 q

For a+1 =5, q = a-1 or for a-1=5 and a+1 = q, it is possible.

So a^2 can be in the form of 5q+1.

Similarly a^2 = 5q can be perfect square for q = 5.

Next for a^2 = 5 q + 4.

(a-2)(a+2) = 5 q

It is possible for a-2 = 5 and a+2=q, or vice versa.

any integer can be in the form of 5q, 5q+1, 5q+2, 5q+3, 5q+4.

For a^2 - 2 = 5q , or for a^2 = 5q +3 , q will be a fraction and not integer.

So a square cannot be in the form of 5q+2 or 5q+3.