What value of $k$ makes the positive five-digit integer $k3,\!57k$ divisible by 18?
To be divisible by 18, a number must be divisible by both 9 and 2. To be divisible by 2, a number must end in an even digit. So $k$ must be either 0, 2, 4, 6, or 8. To be divisible by 9, a number's digits must add up to a multiple of 9. So $k + 3 + 5 + 7 + k$ must be a multiple of 9. Simplifying, I get $2k + 15$ must be a multiple of 9. Since $k$ is even, $2k$ is also even. So $2k + 15$ is odd. The only odd multiples of 9 are 9, 27, 45, 63, etc. I can try each of these and see which one gives me a valid value of $k$. If $2k + 15 = 9$, then $k = -3$, which is not a positive digit. If $2k + 15 = 27$, then $k = 6$, which works. I can check if this is the only solution by trying the next odd multiple of 9. If $2k + 15 = 45$, then $k = 15$, which is not a single digit. So the only value of $k$ that makes $k3,\!57k$ divisible by 18 is 6. 