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Deepak Bora
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Use Euclid Division Lemma to show that the square of any positive integer is either of the form 3q or 3q + 1 for some integer q.

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CBSE 10th Class Mathematics
Past year question paper
Year 2020
SET 1, Code Number  – 30/5/1
Question Number –  28
Euclid Division Lemma
square of any positive integer
Euclid Division Lemma  problem

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  1. Let ‘a’ be any positive integer and b = 3, if a is divided by b by EDL,
    a = 3m + r, m is any positive integer and 0 < r < 3

    If r = 0, a = 3m
    a² = (3m)² = 3 × 3m²
    a² = 3q,

    where 3𝑚² = 𝑞
    r = 1, a = 3m + 1
    a² = (3m + 1)²

    = 9m² + 6m + 1
    = 3 (3m² + 2m) + 1
    a² = 3q + 1

    where q = 3m² + 2m
    r = 2, a = 3m + 2
    a² = (3m + 2)²

    = 9m² + 12m + 4
    = 9m² + 12m + 3 + 1
    = 3 (3m² + 4m + 1) + 1
    a² = 3q + 1,

    where q = 3m² + 4m + 1
    The square of any positive integer is of the form 3q or 3q + 1 for some integer q.

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