Remainders and Exponents

A GRE problem might ask you to solve a Remainder and Exponents problem, where a very large exponent is in play. You can see three examples below:

Examples

How to Solve

It depends on what you're dividing by:

$\div \ 1$: The remainder is always $0$ becuase every integer is divisible by $1$.
$\div \ 2$: The remainder is either $0$ (if the number being divided is even) or $1$ (if the number being divided is odd).
$\div \ 3$: Write down the first one to five remainders and see if you can find some kind of pattern. For example, what is the remainder of $7^{35} \div 3$?
- Remainder of $7^1 \div 3 = 1$
- Remainder of $7^2 \div 3 = 1$
- Remainder of $7^3 \div 3 = 1$
- Ahh, so we can see the remainder must be $1$.
$\div \ 4$: Calculate the final two digits of the number being divided and use your divisibility rule with $4$ to determine the remainder.
$\div \ 5$: Calculate the unit digit of the number being divided. If the unit digit $0$, $1$, $2$, $3$, or $4$, that's the remainder. If the unit digit is $5$, $6$, $7$, $8$, or $9$, the remainder is equal to the unit digit minus $5$.
$\div \ 6$: Try to find some kind of pattern (like in the $\div \ 3$ case).
$\div \ 7$: Try to find some kind of pattern (like in the $\div \ 3$ case).
$\div \ 8$: Calculate the final three digits of the number being divided and use your divisibility rule with $8$ to determine the remainder.
$\div \ 9$: Try to find some kind of pattern (like in the $\div \ 3$ case).
$\div \ 10$: Simply calculate the unit digit of the number being divided. That's the remainder.