Congruency of Numbers: Last digit of $(1234)^{64} + 3,333,333,333,333$ What is a single digit number that is congruent to (1234)$^{64}$ + 3,333,333,333,333 (mod 10)? Show all work.--prophetes.ai

Congruency of Numbers: Last digit of $(1234)^{64} + 3,333,333,333,333$ What is a single digit number that is congruent to (1234)$^{64}$ + 3,333,333,333,333 (mod 10)? Show all work.

HINTS: We can immediately reduce this problem to $(1234)^{64} + 3 \mod 10$ (Can you explain why?)

Then the problem is really to reduce $(1234)^{64} \mod 10$. But this is actually just $4^{64}\mod 10$. Can you see why this one happens? Can you finish from here?