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.
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.