elementary number theory - Working out modulo without a calculator

How does one determine modulo without a calculator in cases like this:
$$15^7 - 13^5(\mod14)$$

Normally I would simply divide what is given by the modulo number and take the decimal output and times it by the modulo number. How can I work out $15^7 - 13^5(mod14)$ without the use of a calculator?

Now what I am thinking is:
$$15 \cong 1 \mod 14 $$
$$15^7 \cong 1 \mod 14$$
$$13 \cong -1 \mod 14$$
$$13^2 \cong 1 \mod 14$$
$$13^5 \cong -1 \mod 14$$
$$[15^7 - 13^5(\mod 14)] = 1 (\mod 14) + 1 (\mod 14) = 2\mod 14$$
Is that right?

Answer

Yes, that's correct! $$[15^7 - 13^5]\pmod{14} \;\;= \;\;2\pmod {14}$$