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Question

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 = aq+r, 0r<a
So, this must be in the form 3q, 3q + 1 or 3q + 2.
Now, (3q)2=9q2=3×3q2=3m where m=3q2
and (3q+1)2=9q2+6q+1
= 3(3q2+2q)+1=3m+1 [where, m=3q2+2q]
Also, (3q+2)2=9q2+12q+4
= 9q2+12q+3+1
3(3q2+4q+1)+1
= 3m + 1 [here, m=3q2+4q+1]
Hence, square of a positive integer of the form 3q + 1 is always in the form 3m + 1 for some integer m.


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