$ 7^{50} \cdot 4^{102} ≡ x \pmod {110} $ The way I would solve this would be: $$ (7^3)^{15} \cdot 7^5 \cdot (4^4)^{25} \cdot 4^2 $$ and take it from there, but I know that this is most likely in an inefficient way. Does anyone have more efficient methods?

Do prime factorization on $110$. It's $110 = 5 \cdot 2 \cdot 11$.

Now work with the prime factors as moduli.

$$7^{50} \cdot 4^{102} \equiv 0 \pmod 2 \text{ trivial, as 2 \mid 4}$$

Then apply Fermat's Little Theorem so we have:

$$7^4 \equiv 1 \pmod 5 \implies 7^{50} \equiv 49 \equiv 4 \pmod 5$$ $$4^4 \equiv 1 \pmod 5 \implies 4^{102} \equiv 4^2 \equiv 1 \pmod 5$$

Now multiply them and we have:

$$7^{50} \cdot 4^{102} \equiv 4 \cdot 1 \equiv 4 \pmod 5$$

Now repeat the method with the last factor:

$$7^{10} \equiv 1 \pmod {11} \implies 7^{50} \equiv 1 \pmod {11}$$ $$4^{10} \equiv 1 \pmod {11} \implies 4^{102} \equiv 4^2 \equiv 16 \equiv 5 \pmod {11}$$

Multiply them and we have:

$$7^{50} \cdot 4^{102} \equiv 1 \cdot 5 \equiv 5 \pmod {11}$$

Now just apply CRT to the three congruence relation to get the final answer.