Number Theory

Foundational

The study of integers, and specifically of divisibility — which numbers divide which, what's left over when one doesn't, and how factorizations constrain the answer. A surprising number of AMC 10 problems that don't mention "number theory" at all are secretly asking a divisibility or remainder question underneath their surface framing.

Why this domain matters on the AMC 10

Number theory problems appear throughout the exam, not clustered at one difficulty level — easy problems test straightforward divisibility, while hard problems disguise a modular-arithmetic or Diophantine argument behind an innocent-looking setup. The domain rewards pattern recognition (spotting that a problem is really about remainders, or about counting divisors) more than raw computation, and several of its tools — modular arithmetic especially — are also the fastest route to answers in problems from other domains that happen to involve integers.

Prerequisite path

Number Theory builds on Algebra Foundations — comfort with basic algebraic manipulation and exponent rules is assumed throughout.

Algebra Foundations Number Theory Counting & Probability Geometry Mixed Problem Solving Hard AMC 10 Early AIME

Modules

Lessons are ordered; each builds on the last. Work through them in sequence the first time, then revisit individual lessons as needed during review.

Key formulas & techniques

GCD, LCM, and the relationship between them

gcd(a,b) × lcm(a,b) = a × b

The Euclidean algorithm computes gcd(a,b) in a handful of steps regardless of how large a and b are — far faster than factoring both numbers first.

Modular arithmetic

a ≡ b (mod m) means m divides (a − b)

Addition, subtraction, and multiplication all behave predictably under a fixed modulus — this is what makes units-digit and remainder problems tractable without computing the full (often enormous) value.

Counting divisors

If n = p₁^e₁ p₂^e₂ ⋯ pₖ^eₖ, the number of divisors is (e₁+1)(e₂+1)⋯(eₖ+1)

Every divisor-counting question reduces to prime factorization first — this formula is the payoff for doing that factorization correctly.

Common traps

Confusing "divides" direction

"a divides b" (written a | b) means b is a multiple of a, not the other way around. Misreading this direction silently inverts an entire argument.

Forgetting that remainders must be nonnegative and less than the modulus

A remainder is always in the range 0 to m−1 for modulus m. A negative intermediate result needs its remainder adjusted back into that range, not reported directly.

Applying a divisor-counting or sum-of-divisors formula before fully factoring

These formulas require the complete prime factorization. A partial factorization (stopping at a composite factor) gives a wrong answer, not just an incomplete one.

Curated resources

Unit test

All 8 lessons in this domain are built. A 16-question cumulative assessment — two questions per lesson, presented in mixed order rather than grouped by topic — is the honest way to check whether the material holds up once nothing is telling you which lesson a problem came from.

Take the unit test

Domain mastery checklist

Check these off only once true, not as encouragement — each should be genuinely automatic before moving on to Counting & Probability or Geometry.