Dear Uncle Colin,

I’ve been told that $(ax+b)(bx+a) \equiv 10x^2 + cx + 10$, with $a$ and $b$ positive integers, and I need to find the possible values for $c$. How?!

- This Ridiculous Identity Puzzle Looks Easy

Hi, TRIPLE, and thanks for your message!

The first thing I would do would be to expand the left hand side and try to match coefficients.

• $abx^2 + (a^2 + b^2)x + ab \equiv$10x^2 + cx + 10$That means that$ab =10$and$a^2 + b^2 = c$. That’s not enough to solve for$c$in the reals – we have two equations in three unknowns – but there aren’t many possibilities for$a$and$b$in the positive integers. There are only two possible pairs of answers for$a$and$b$: they’re either$1$and$10$in either order, or they’re$2$and$5$in either order. If they’re$1$and$10$, then$c = 1^2 + 100^2 = 101$. If they’re$2$and$5$, then$c = 2^2 + 5^2 = 29$. So the possible values for$c\$ are 101 and 29.

Hope that helps!

- Uncle Colin