Question

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Answer 1

To Prove: that the square of a positive integer of the form 5q + 1 is of the same form

Proof: Since positive integer n is of the form 5q + 1

If n = 5q + 1

$$\begin{gathered} \mathrm{Then}{n}^2={\left(5q+1\right)}^2 \\ \Rightarrow n^2={\left(5q\right)}^2+{\left(1\right)}^2+2\left(5q\right)\left(1\right) \\ \Rightarrow n^2=25q^2+1+10q \\ \Rightarrow n^2=25q^2+10q+1 \\ \Rightarrow n^2=5\left(5q^2+2q\right)+1 \end{gathered}$$

Hence n² integer is of the form 5m + 1.