### Instruction: What is the smallest base-10 integer that can be represented as $AA_5$ and $BB_7$, where $A$ and $B$ are valid digits in their respective bases?### Response: So the question is asking for the smallest integer that can be expressed as $AA_5$ and $BB_7$?
Yes, let's call that integer $n$.
Well, we know that $n = A\cdot5^1 + A\cdot5^0 = B\cdot7^1 + B\cdot7^0$.
So we can write $n = 5A + A = 7B + B$.
That simplifies to $n = 6A = 8B$.
So, $n$ must be a multiple of 8.
And it must also be a multiple of 6.
So $n$ must be a multiple of their least common multiple, which is 24.
So the smallest possible value of $n$ is 24.