NUMBER THEORY
Divisibility, primes, modular arithmetic, GCD/LCM, digits, and bases.
What Number Theory means on the AMC10
Number theory questions are about the structure of the integers. The most common tools:
- Divisibility and primes — divisibility tests, prime factorization
- GCD and LCM — the relationship
- Modular arithmetic — solving "what is the remainder when..." problems quickly
- Digit problems — units digit, sum of digits, palindromes
- Number bases — converting and reasoning in non-base-10
A surprising fraction of these problems collapse to mod 9 or mod 10 tricks once you spot the pattern.
SUBTOPICS
Techniques for exploiting the decimal representation of integers — digit sums, reversals, counting digit-constrained numbers, trailing zeros, and units-digit patterns.
Finding integer solutions to equations using divisibility, factoring tricks, and bounding arguments.
Quick tests for divisibility, prime factorization, and counting divisors.
Greatest common divisor and least common multiple — tools for divisibility, cycles, and scheduling problems on the AMC10.
Reasoning about remainders to solve divisibility, units-digit, and large-exponent problems without heavy computation.
Understanding place value in any base b, converting numbers between bases, and solving equations where the base itself is unknown.