KBB3 Problem 1

Find the smallest prime number that divides $54^{54} + 55^{55} + 56^{56}$

Solution

The number is odd, so it's not divisible by 2. It's also not divisible by either 3 nor 5 because

$\displaystyle 54^{54} + 55^{55} + 56^{56} \equiv 0 + 1^{55} + (-1)^{56} \equiv 2 \mod 3$

$\displaystyle 54^{54} + 55^{55} + 56^{56} \equiv (-1)^{54} + 0 + 1^{56} \equiv 2 \mod 5$

But it is divisible by 7 because:

$\displaystyle 54^{54} + 55^{55} + 56^{56} \equiv (-2)^54 + (-1)^{55} +0 \equiv (-8)^{18} -1 \equiv (-1)^{18}-1 \equiv 0 \mod 7$