2021 April MIMC 10 Problems/Problem 9

Find the largest number in the choices that divides $11^{11}+13^2+126$.

$\textbf{(A)} ~1 \qquad\textbf{(B)} ~2 \qquad\textbf{(C)} ~4 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~16$

Solution

We can look at the last digit of the expression first. $11^n\equiv1$ (mod $10)$ and $13^2\equiv9$ (mod $10$). Therefore, the expression $11^{11}+13^2+126\equiv1+9+6\equiv6$ (mod $10$). We know that it is divisible by $2$ at this point. Then, we can look at the last two digits. $11^{11}\equiv11$ (mod $100$) and $13^2\equiv69$ (mod $100$). The expression $11^{11}+13^2+126\equiv11+69+6\equiv86$ (mod $100$) $\equiv2$ (mod $4$). Therefore, our answer is $\fbox{\textbf{(B)} 2}$.

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