Modular Arithmetic with Sines Given $$\sin(10^{100})+\sin(n)=0$$ find $n$.
I wrote so far that $$\sin(10^{100})=\sin(10^{100} \mod 360)$$ and I noticed that $10^3 \mod 360=280$ and $10^4 \mod 360=280$ so I (correctly) assumed that $$10^{100} \mod 360=280$$
but why is this the case?

It seems your argument is in degrees, as you are worried with residue modulo 360, so I will make this assumption. Modular rings work like a cartesian product of the modulos of the prime factor powers. So, $\mathbb{Z}_{360}$ behaves exactly like $\mathbb{Z}_8\times\mathbb{Z}_9\times\mathbb{Z}_5$. Writing 10 in the latter ring, we have $2\times 1\times 0$. Then cube that to get $10^3$: $0\times 1\times 0$. Multiplying this (component-wise) by 10 ($2\times 1\times 0$) does not change it. So, any power of 10 three and above is equal to any other.
