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

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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.

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