AMC 10 Number Theory — Paper 2
Premium AMC 10 number theory paper: 30 hard problems on Legendre valuations, multiplicative order, totients and Chinese remainder, with worked solutions.
About this test
AMC 10 Number Theory — Paper 2 is a full 30-question paper you attempt online in 75 minutes. It follows the American Mathematics Competition 10 pattern, so the timing, the question style and the marking match the real exam. There is no negative marking, so it is worth attempting every question. Leaving a question blank still earns 1.5 marks.
You get your score the moment you submit, question by question, with a step-by-step solution to every problem. 25 questions in 75 minutes. A high score qualifies for the AIME.
What it covers
Sections
- Divisor counting — 2 questions
- Modular exponentiation — 2 questions
- Euler totient — 1 question
- Legendre valuation — 1 question
- Chinese remainder — 1 question
- Multiplicative order — 1 question
- Units digits — 1 question
- Coprime factorisations — 1 question
- Kummer / carries — 1 question
- Harshad numbers — 1 question
- Sum of divisors — 1 question
- Modular inverse — 1 question
- Twin primes — 1 question
- Highly composite — 1 question
- Fermat little theorem — 1 question
- Linear Diophantine — 1 question
- Perfect squares — 1 question
- Central binomial — 1 question
- Digit patterns — 1 question
- Self-inverse residues — 1 question
- GCD sums — 1 question
- Triangular numbers — 1 question
- Power towers — 1 question
- Inclusion-exclusion — 1 question
- Trailing zeros — 1 question
- Pythagorean triples — 1 question
- Totient sums — 1 question
- Abundant numbers — 1 question
Difficulty spread
- 1 easy question
- 7 medium questions
- 9 hard questions
Before you start
15 questions · 45 minutes.
- Each correct answer scores 6 points.
- Each unanswered question scores 1.5 points.
- A wrong answer scores 0 — a blank is worth more than a guess unless you can eliminate at least three options.
- No calculators. Rough work on paper.
This paper covers Number Theory only, so a weak score points at one topic rather than leaving you guessing.
A look at the questions
Questions people ask
How does this paper differ from Number Theory Paper 1?
It is the premium tier. Most questions combine two ideas rather than one — Legendre with Kummer, totients with gcd sums, Chinese remainder with order arguments — which is where AMC 10 questions 20 to 25 live.
What topics does it cover?
Divisor functions, Legendre valuations, Kummer's theorem on binomial coefficients, multiplicative order, Euler's totient and totient sums, Chinese remainder problems, Harshad numbers, abundant numbers and inclusion-exclusion.
How is it scored?
Real AMC 10 rules: 6 points for a correct answer, 1.5 for a blank, 0 for a wrong one. On a paper this hard the blank rule genuinely matters — a wild guess has negative expected value.
Do I need Paper 1 first?
It helps but is not required. Paper 1 covers the same topics at contest-entry level; this one assumes you already know what a totient and a valuation are.