Question

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

Answer 1

let {a}be the square of an integer and b=5.applying euclids division lemma [a=5q+2,for some integer and r=2 or 3.so a=5q+2 or 5q+3.this topic is from real numbers.actually we are having one formula a=bq+r.so it is called euclids division algorithm.this result was first recorded in book 7.
i hope it is useful

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.