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

Solution:
Let a be the positive integer and b = 5.
Then, by Euclid’s algorithm, a = 5m + r for some integer m ≥ 0 and r = 0, 1, 2, 3, 4 because 0 ≤ r < 5.
So, a = 5m or 5m + 1 or 5m + 2 or 5m + 3 or 5m + 4.
So, (5m)² = 25m² = 5(5m²)
= 5q, where q is any integer.
(5m + 1)² = 25m² + 10m + 1
= 5(5m² + 2m) + 1
= 5q + 1, where q is any integer.
(5m + 2)² = 25m² + 20m + 4
= 5(5m² + 4m) + 4
= 5q + 4, where q is any integer.
(5m + 3)² = 25m² + 30m + 9
= 5(5m² + 6m + 1) + 4
= 5q + 4, where q is any integer.
(5m + 4)² = 25m² + 40m + 16
= 5(5m² + 8m + 3) + 1
= 5q + 1, where q is any integer.
Hence, The square of any positive integer is of the form 5q, 5q + 1, 5q + 4 and cannot be of the form 5q + 2 or 5q + 3 for any integer q.