Question 5
A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1. i.e., 3m or 3m +2 for some integer m? Justify your answer.

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Solution

No, we cannot.
By Euclid's lemma, $b = a q + r, 0 \leq r < 3$
So, this must be in the form 3q, 3q + 1 or 3q + 2.
Now, $(3 q)^{2} = 9 q^{2} = 3 m$
and $(3 q + 1)^{2} = 9 q^{2} + 6 q + 1$
$= 3 (3 q^{2} + 2 q) + 1 = 3 m + 1$ $[w h e r e, m = 3 q^{2} + 2 q]$
Also, $(3 q + 2)^{2} = 9 q^{2} + 12 q + 4$
$= 9 q^{2} + 12 q + 3 + 1$
$3 (3 q^{2} + 4 q + 1) + 1$
= 3m + 1 $[h e r e, m = 3 q^{2} + 4 q + 1]$
Hence, square of a positive integer is of the form 3q + 1 is always in the form 3m + 1 for some integer m.

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A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, 3m or 3m + 2 for some integer m? Justify your answer.

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Question 4
Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

Write whether the square of any positive integer can be of the form 3m+2 , where m is a natural number . Justify your answer ?

Q. 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.]

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Question 5 A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1. i.e., 3m or 3m +2 for some integer m? Justify your answer.

Question 5
A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1. i.e., 3m or 3m +2 for some integer m? Justify your answer.