Remainders are fun

We note that $\space 28^4 = 614656 \equiv 1 \pmod{45}$ .

$\Rightarrow 28^{45} \equiv 28^{4\times 10+1} \pmod{45} \equiv 28 \left( 28^4 \right)^{10} \pmod{45} \equiv \boxed{28} \pmod{45}$

$28^2 \equiv 19 \pmod{45}$

$28^3 \equiv 37 \pmod{45}$

$28^5 \equiv 28 \pmod{45}$ from first two.

$28^{10} \equiv 19 \pmod{45}$

$28^{20} \equiv 1 \pmod{45}$

$28^{40} \equiv 1 \pmod{45}$

$28^{45} \equiv 28 \pmod{45}$ from equation (3) and (6)

hence the answer is $\boxed{28}$

Since $gcd(28, 45)=1$ , we can use Euler's Theorem . We know that $\phi(45)=24$ .

${28}^{45}\equiv(28)^{2\times{\phi(45)}}\times{28}\equiv1\times{28}\equiv28\pmod{45}$