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
Deepak BoraNewbie
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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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.