2025 AMC 10A Problem 17

Both $273436 - 16$ and $272760 - 15$ are multiples of $N$

So is their difference; reduce it by subtracting further multiples, and remember $N \gt 16$ since the remainder is $16$

Subtract off the remainders. Both $273420 = 273436 - 16$ and $272745 = 272760 - 15$ are multiples of $N,$ so their difference $675$ is too. Now $272745 = 404 \cdot 675 + 45,$ which makes $45$ a multiple of $N$ as well. The remainder $16$ means $N \gt 16,$ and the only divisor of $45$ bigger than $16$ is $45$ itself. So $N = 45,$ and its tens digit is $4.$ Thus, E is the correct answer.