### Instruction: What is the remainder when $2^{87} +3$ is divided by $7$?### Response: I want to find a way to simplify the expression $2^{87} +3$ so that it is easier to work with modulo $7$.
I notice that $2^3 = 8 \equiv 1 \pmod 7$, so I can use this fact to reduce the exponent of $2$.
If I divide $87$ by $3$, I get $29$ with a remainder of $0$, so $2^{87} = (2^3)^{29} \equiv 1^{29} \equiv 1 \pmod 7$.
This means that $2^{87} +3 \equiv 1 + 3 \equiv 4 \pmod 7$.
So the remainder when $2^{87} +3$ is divided by $7$ is $4$.