Use Euclid’s division lemma two show that the square of any positive integer is either of the forms or for some integer

Question – Use Euclid’s division lemma two show that the square of any positive integer is either of the forms or for some integer.

Solution:

Let x be any positive integer, which gives q as the quotient and r is the remainder when it is divided by 3.
Then using Euclid’s division lemma a = BQ + r, 0 ≤ r < b we get,
a = 3q + r, 0 ≤ r < 3,
So possible values of r are 0, 1, 2.
Now, when r = 0, x = 3q
Therefore, x²  = (3q)²  = 9q² = 3 x 3q² = 3m, where m= 3q²
When, r = 1, x = 3q + 1
x² = (3q + 1)² = 9q² + 6q + 1 = 3(3q² + 2q) + 1 = 3m + 1 where m = 3q² + 2q
And when r = 2, x = 3q + 2
Hence x² = (3q + 2)²
= 9q² + 12q + 4
= 9q² + 12q + 3 + 1
= 3(3q² + 4q + 1) + 1
= 3m + 1, where m= 3q² + 4q + 1) + 1
Therefore, the square of any positive integer is either of form 3m or 3m + 1 for some integer m.

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