GRE-Style: Remainder Problems

GRE Quantitative · Axiom Academy

EXAMPLE GRE-Style: Remainder Problems Spotting a repeating remainder cycle to solve a power-of-2 remainder problem What is the remainder when 2^ 10 is divided by 5? Nice work — you used a remainder cycle to avoid ever computing 2^ 10 directly. Look for a cycle: the remainders of repeat every 4 powers: . This is much faster than computing large powers directly. Use the cycle length as a clock: since , the remainder for n=10 matches the remainder for n=2 — the 2nd spot in the cycle. Remainders add and multiply directly: if and , then , and similarly — so gives . You never need the full product to find its remainder. Watch for remainders in word problems: "how many are left over after filling boxes/groups" is a remainder question in disguise — identify the total and the group size, then divide. Cyclicity is one of the highest-leverage GRE number-properties tricks: whenever a problem asks for a remainder involving a large exponent, look for the repeating pattern before you reach for a calculator.

This is the written version of the interactive lesson above. See the full GRE Quantitative course.