Question 4

Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.
[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

Answer

Let x be any positive integer and b = 3.

Then, by euclid’s algorithm

x = 3q + r, where q ≥ 0 and r = 0, 1, 2 [0 ≤ r ≤ b]

Case (i) : For r = 0, x = 3q, = x² = 9q², taking 3 as common,

x² = 9q² = 3 (3q²), which is of the form 3m, where m = 3q².

Case (ii) : For r = 1, x = 3q + 1

x² = 9q² + 1 + 6q, taking 3 as common,

= 3 (3q² + 2q) + 1, which is of the form 3m + 1, where m = 3q² + 2q

Case (iii) : For r = 2, 3q + 2

x² = 9q² + 4 + 12q = (9q² + 12q + 3) + 1, taking 3 as common,

= 3 (3q² + 4q + 1) + 1, which is of the form 3m +1, where m = 3q² + 4q + 1

Hence, x² is either of the form 3m, 3m + 1 for some integer m.

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