AMC 10 Number Theory — Paper 1
Free AMC 10 Number Theory practice paper: 30 questions at real contest difficulty, 75 minutes, with full worked solutions for every problem.
About this test
AMC 10 Number Theory — Paper 1 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 arithmetic — 3 questions
- Digit problems — 2 questions
- GCD — 1 question
- Trailing zeros — 1 question
- Base representation — 2 questions
- Primes — 1 question
- Sum of divisors — 1 question
- Legendre valuation — 2 questions
- Order modulo n — 1 question
- Diophantine — 3 questions
- Chinese remainder — 1 question
- Digit sums — 1 question
- Perfect squares — 1 question
- LCM — 1 question
- Units digits — 1 question
- Divisor sums — 1 question
- Modular inverse — 1 question
- Consecutive integers — 1 question
- Fermat — 1 question
- Perfect numbers — 1 question
- Order and primitive roots — 1 question
Difficulty spread
- 5 easy questions
- 17 medium questions
- 8 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 hard is this AMC 10 number theory paper?
It is pitched at the real contest, not at practice-book level. The first eight questions are approachable, the middle fourteen are the band that decides most AMC 10 scores, and the last eight sit at the difficulty that separates AIME qualifiers.
What topics does it cover?
Divisor counting and sums, prime factorisation, GCD and LCM, modular arithmetic and orders, Chinese remainder problems, digit and base representation, Legendre valuations, and Diophantine equations.
How is it scored?
Exactly like the real AMC 10: 6 points for a correct answer, 1.5 for a blank, 0 for a wrong one. On a hard paper the blank rule matters — a wild guess costs you 1.5 points, so leave it unless you can eliminate three options.
How long should it take?
75 minutes for 30 questions. That is deliberately tighter than three minutes each, because pace under pressure is part of what the AMC tests.