Question

# 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, 0≤r<b, here a is any positive integer, When we take the divisor (b), we have a=3q+r for 0≤r<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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