For each integer a if a2-1 is even then 4 divides a2-1?

Answer 1

Assuming it's a2-1 you mean.

If a2-1 is even, it can be expressed as a2-1 = 2 * t, where t is a natural number.

a2-1 = (a - 1)(a + 1) = 2 * t

You immediately see that at least one of the terms in parenthesis has to be even, and then, that actually both have to be even, because (a + 1) - (a - 1) = 2.

Therefore, you can express both of the terms as:

a - 1 = 2 * s

a + 1 = 2 *r,

which gives us:

2 * s * 2 * r = a2 - 1, simplify:

4 * r * s = a2 - 1.

Therefore, if a2-1 is even, a2-1 is also divisible by 4.