I found this to be an interesting mathematical one. Apart from the $ marks, it gets off to a good start, but it messes up on the final formula for ab and what follows.

Using their notation, the formula for ab has the two, usually different, solutions:

ab = (xp – yq)^{2 + (xq + yp)}2 and

ab = (xp +yq)^{2 + (xq-yp)}2

Their solution of ab = (xp+yq)^{2 + (xq+yp)}2 does not work., but they had the right idea.

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Use a formula to show that if two whole numbers are the sum of two squares, so is their product

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If two whole numbers $a$ and $b$ can be written as the sum of two squares, that is, $a = x^{2 + y}2$ and $b = p^{2 + q}2$ for some whole numbers $x$, $y$, $p$, and $q$, then their product $ab$ can also be written as the sum of two squares. Specifically, we have

$$ab = (x^{2 + y}2)(p^{2 + q}2) = x^{2 p}2 + y^{2 p}2 + x^{2 q}2 + y^{2 q}2$$

By the commutative and associative properties of multiplication, this expression can be rearranged to give the two, usually different solutions:

$$ab = (x^{2 p}2 + y^{2 q}2) + (x^{2 q}2 + y^{2 p}2) = (xp + yq)^{2 + (xq + yp)}2$$

Thus, $ab$ can be written as the sum of two squares.

For example, let $a = 12$ and $b = 5$. We can write $12 = 4^{2 + 2}2$ and $5 = 2^{2 + 1}2$, so

$$ab = (4^{2 + 2}2)(2^{2 + 1}2) = (4 \cdot 2 + 2 \cdot 1)^{2 + (4 \cdot 1 + 2 \cdot 2)}2 = 14^{2 + 10}2 = \boxed{196}$$