The Remainder Detective

Questions like 'What is the remainder when X is divided by Y?' — especially when X is huge (a big exponent) or contains a variable.

Trying to compute the actual value. 3^64 is a 31-digit number. You never need it — you need its remainder.

1. Test small cases first (n=1, 2, 3, 4, 5) and write out the remainders.
2. Look for the cycle. Most remainders repeat every 2, 3, 4, or 5 cases.
3. Reduce the problem using the cycle length: e.g., '7^100 mod 4' becomes 'find where 100 lands in the cycle.'
4. Divisibility shortcuts to know cold: a number is divisible by 3 if its digit sum is divisible by 3. Divisible by 4 if last two digits form a multiple of 4. Divisible by 5 if last digit is 0 or 5. Divisible by 9 if digit sum is divisible by 9.
5. If the question is QC and remainders vary across cases, the answer is D (cannot be determined).

This is THE most-tested concept on hard GRE Quant. Master it and you crack 15-20% of the section.