Question

Suppose that a is an integer number. Show that if a^2 is divisble by 5, then a is divisble by 5.
Hint: A integer divisble by 5 if its last digit is either 0 or 5

Answer 1

By contraposition:
Let k be an integer that is not divisible by 5.
\[\implies k = 5q + r \text{, such that } q \in Z \text{ and } r \in\{1,2,3,4\}\]
\[\implies k^2 = (5q+r)^2 = 25q^2 + 10qr + r^2 = 5(5q^2 + 2qr) + r^2\]
\(\implies k^2 \) is not divisable by 5. QED.

Answer 2

ok thank you