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, $a = x2 + y2$ and $b = p2 + q2$ 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
$$ab = (x2 + y2)(p2 + q2) = x2 p2 + y2 p2 + x2 q2 + y2 q2$$
By the commutative and associative properties of multiplication, this expression can be rearranged to give the two, usually different solutions:
$$ab = (x2 p2 + y2 q2) + (x2 q2 + y2 p2) = (xp + yq)2 + (xq + yp)2$$
Thus, $ab$ can be written as the sum of two squares.
For example, let $a = 12$ and $b = 5$. We can write $12 = 42 + 22$ and $5 = 22 + 12$, so
$$ab = (42 + 22)(22 + 12) = (4 \cdot 2 + 2 \cdot 1)2 + (4 \cdot 1 + 2 \cdot 2)2 = 142 + 102 = \boxed{196}$$