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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.


Solution

No, we cannot express the square of any positive integer can be of the form 3m + 2.
By Euclid's lemma,
a=bq+r, 0r<b, here a is any positive integer,

When we take the divisor (b), we have

a=3q+r for 0r<3


So, any positive integer is of the form 3q, 3q + 1, 3q + 2.
Now, (3q)2=9q2=3m (where m=3q2)

(3q+1)2=9q2+6q+1=3(3q2+2q)+1=3m+1 (where m=3q2+2q

= (3q+2)2=9q2+12q+4=9q2+12q+3+1=3(3q2+4q+1)+1=3m+1 (where m=3q2+4q+1)

Thus, any square is of the form 3m or 3m + 1.


Hence, square of any positive number cannot be of the form 3m + 2.


Mathematics

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