a) $1082^{551} \equiv 1(\bmod 47)$
b) $1081^{552} \equiv 1(\bmod 47)$
c) $1080^{551} \equiv -1(\bmod 47)$
d) $1079^{553} \equiv 1079(\bmod 47)$
You have to use Fermat's little theorem which states that ....
if $p$ is a prime number and $a$ is a whole number such that $p \nmid a$, then
$$a^{p-1} \equiv 1(\bmod p)$$
From that I can immediately conclude that congruence $b)$ is not valid since $1081/47 = 23$
As for congruence $a)$ I use the fact that $551 = 46*11 + 45$ and from Fermat's little theorem get that
$$1082^{45} \equiv 1(\bmod 47) $$
But this is also too large of a number to evaluate, is there quick and clever way to check all these congruences?
for a) this is right for b),c) we have $1081=23\cdot 47$ so the given result is wrong, i must be $0$ d) $$1079^{553}=1079\equiv 45\mod 47$$