$222^{333} + 333^{222}$ is divisible by which of the following numbers?
$222^{333} + 333^{222}$ is divisible by which of the following numbers?
2 and 3 but not 37
3 and 37 but not 2
2 and 37 but not 3
2, 3 and 37
Solution
Since $222^{333}$ is even and $333^{222}$ is odd, their sum is odd, so it is not divisible by 2. Writing $222 = 2 \times 3 \times 37$ and $333 = 9 \times 37 = 3^2 \times 37$, both terms share factors of 3 and 37. The expression can be factored as $111^{222} \times [\text{odd term}]$, and since $111 = 37 \times 3$, the sum is divisible by 3 and 37 but not 2.