### Instruction: Find the positive base $b$ in which the equation $5 \cdot 25_b = 137_b$ is valid.### Response: I notice that the equation has both multiplication and equality in base $b$, which means that I need to convert the digits into their decimal equivalents using powers of $b$.
For example, $25_b = 2b + 5$ and $137_b = b^2 + 3b + 7$.
So the equation becomes $(5)(2b + 5) = b^2 + 3b + 7$.
I can expand the left side and simplify the equation to get $10b + 25 = b^2 + 3b + 7$.
Subtracting $10b + 25$ from both sides, I get $b^2 - 7b - 18 = 0$.
This is a quadratic equation that I can try to factor.
I look for two numbers that multiply to -18 and add to -7.
I see that -9 and 2 work, so I can write $b^2 - 7b - 18 = (b - 9)(b + 2)$.
This means that $b = 9$ or $b = -2$.
However, $b$ has to be a positive base, so I can ignore the negative option.