Find the Smallest Prime Factor of the Quotient
==============================================
Another one from YouTube I want to solve without watching the video by
the great mathematician Michael Penn: find the smallest prime factor of

 2022²⁰²² + 6
 ------------
      6

2022 is divisible by 6, and

  2022
  ---- = 337
    6

Great!

Now,

  2022²⁰²² = 2022×2022²⁰²ⁱ = 6×337×2022²⁰²ⁱ

which means that

  2022²⁰²² + 6
  ------------ = 337×2022²⁰²ⁱ + 1
        6

2 and 3 are not factors of this number because 2022 is a multiple of 6, thus

  337×2022²⁰²ⁱ + 1 ≡ 1 mod 2

and

  337×2022²⁰²ⁱ + 1 ≡ 1 mod 3

The next prime number is 5, and by Fermat's little theorem, if p is a prime
and a is a natural number not divisible by p:

  aᵖ⁻ⁱ ≡ 1 mod p

and

  2021 = 2020 + 1 = 4 × 505 + 1 

which means that

  2022²⁰²ⁱ = 2022⁴ˣ⁵⁰⁵⁺ⁱ = 2022×2022⁴ˣ⁵⁰⁵ ≡ 2022×1⁵⁰⁵ mod 5
           ≡ 2022×1 mod 5 ≡ 2 mod 5

and

  337 = 335 + 2 ≡ 2 mod 5

Thus,

  337×2022²⁰²ⁱ + 1 ≡ 2×2 + 1 mod 5 ≡ 0 mod 5

Great! The smallest prime factor of

 2022²⁰²² + 6
 ------------
      6

is 5