use euclids 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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answer is given below
Step-by-step explanation:
Let 'a' be any positive integer.
On dividing it by 3 , let 'q' be the quotient and 'r' be the remainder.
Such that ,
a = 3q + r , where r = 0 ,1 , 2
When, r = 0
∴ a = 3q
When, r = 1
∴ a = 3q + 1
When, r = 2
∴ a = 3q + 2
When , a = 3q
On squaring both the sides,
When, a = 3q + 1
On squaring both the sides ,
When, a = 3q + 2
On squaring both the sides,
Therefore , the square of any positive integer is either of the form 3m or 3m+1.
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