Question

161) if q is an integer that can be expressed as the sum of two integer squares, show that both 2q and 5q can also be expressed as the sum of two integer squares

Answer 1

Given that q is an integer that can be expressed as the sum of two integer squares, Let's show that 2q and 5q can also be expressed as the sum of two integer squares.

Let a and b represent the integers.

We have:

a² + b² = q

To show that both 2q can also be expressed as the sum of two integer squares, we have:

`\begin{gathered} (a+b)^2+(a-b)^2 \\ \\ =(a+b)(a+b)+(a-b)(a-b) \\ \\ =a(a+b)+b(a+b)+a(a-b)-b(a-b)^{} \\ \\ =a^2+ab+ab+b^2+a^2-ab-ab+b^2 \\ \\ =a^2+a^2+ab+ab-ab-ab+b^2+b^2 \\ \\ =2a^2+2b^2 \\ \\ =2(a^2+b^2) \end{gathered}`

Since a² + b² = q, we have:

2(a² + b²) = 2q

From the above we have shown that 2q can also be expressed as the sum of two integer squares.

• To show that both 5q can also be expressed as the sum of two integer squares, let's apply the expression below:

`\begin{gathered} (a+2b)^2+(2a-b)^2 \\ \\ =a^2+4ab+4b^2+4a^2-4ab+b^2 \\ \\ =a^2+4a^2+4ab-4ab+4b^2+b^2 \\ \\ =5a^2+5b^2 \\ \\ =5(a^2+b^2) \\ \\ =5q \end{gathered}`

Using the above expression, we have shown that 5q can also be expressed as the sum of two integer squares.