Modular maths: How do I find the remainder? How do I find the remainder of $5^{22} \pmod{25}$?
And also how do I find the remainder of $3^{16} + 7 \pmod{5}$?

Try to see which power of $5$ can give $1$ or $-1$ modulo $7$. This will reduce your computations quite significantly. For example, $5^2 \equiv 4 \pmod{7}$ so $5^3 \equiv 20 \equiv 6 \equiv -1 \pmod{7}$. Thus
$$5^{22} \equiv 5^{21} \cdot 5 \equiv (5^{3})^7 \cdot 5 \equiv -5 \equiv 2 \pmod{7}$$
