A positive integer is of the form \(3q+1\), q being a natural number. Can you write its square in any form other than \(3m+1\)? Justify your answer.
No. The square of such a number is always of the form \(3m+1\).
Step 1: Express the number in the given form.
Let the number be \(n = 3q + 1\), where \(q\) is a natural number.
Step 2: Find its square.
\(n^2 = (3q + 1)^2\)
Expand: \(n^2 = 9q^2 + 6q + 1\)
Step 3: Factorize the expression.
\(n^2 = 3(3q^2 + 2q) + 1\)
Here, \(3q^2 + 2q\) is an integer. Let \(m = 3q^2 + 2q\).
Step 4: Final form.
So, \(n^2 = 3m + 1\).
Conclusion. The square of a number of the form \(3q+1\) is always of the form \(3m+1\). It can never be written as \(3m\) or \(3m+2\).