What's the smallest prime factor of [tex: \frac{2022^{2022} + 6}{6}]?

ur mom

The number in question is 337*2022^2021+1
337*2022^2021 mod 10 is 4, meaning 337*2022^2021+1 is 5 mod 10.
337*2022^2021+1 is a succesor of a multiple of 2 and 3, which means it can't be multiples of any of them, therefore it's smallest prime factor is 5.

Or just, you know, put (2022**2022+6)%(6*p) into python and just check.

So, [tex: 337\times 2022^2021 + 1]
The answer is 5. What's the point of such a question when the answer is so low?
[tex: 337\times 2022^2021 + 1 \equiv 2^2022 + 1 \equiv 4^1011 + 1 \equiv 4\times 1^505 + 1 \ equiv 0 (mod 5)]

The same algorithm can be used to find bigger factors:
The prime factors less than 2^32 are (5, 7, 251, 19777).

Finding all the factors would require a BigInt library and probably a super-computer.