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?!
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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
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