Question

use Euclid division lemma to show that the square of any positive integer is either of the form 3m or 3m+1 for some integer m.​

Answer 1

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Answer 2

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By Euclid’s division algorithm

a = bq + r, where 0 ≤ r ≤ b

Put b = 3

a = 3q + r, where 0 ≤ r ≤ 3

If r = 0, then a = 3q

If r = 1, then a = 3q + 1

If r = 2, then a = 3q + 2

Now, (3q)² = 9q2

= 3 × 3q²

= 3m, where m is some integer

(3q + 1)² = (3q)² + 2(3q)(1) + (1)²

= 9q² + 6q + 1

= 3(3q² + 2q) + 1

= 3m + 1, where m is some integer

(3q + 2)² = (3q)² + 2(3q)(2) + (2)²

= 9q² + 12q + 4

= 9q² + 12q + 4

= 3(3q² + 4q + 1) + 1

= 3m + 1, where m is some integer.

Hence the square of any positive integer is of the form 3m, or 3m +1 But not of the form 3m + 2