24 original competition-style problems: divisibility, primes, remainders and digits. Try each one before opening the hints; the second hint gives more away, and the full solution explains why the method works and where the idea leads.
Every question is free to read. The hints, answer checking and full solutions are included with every A Level, IB, IGCSE and CBSE plan. See plans.
Answer a problem to track what you have solved.
Press Project on any problem to show it full screen with a timer, the hints, the answer and the worked solution one step at a time (arrow keys move between problems; Space reveals the next step; F full screen; Esc closes). Switch on the ‘Add to worksheet’ buttons, pick problems, then print them from the worksheet builder or set them as homework for a class, with the full solutions as the mark scheme. Free problems are free for every class; problems marked ‘With a plan’ can be set by teachers with a plan or school licence. Ready-made sessions: maths club packs.
How many positive whole numbers n have the property that n + 1 divides n2 + 1? Prove that you have found them all.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Proof techniques
For how many positive whole numbers n is 2n + 3n a perfect square?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Proof techniques
Find the sum of all primes p for which 2p + p2 is also prime.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Proof techniques
How many ordered pairs (x, y) of positive whole numbers satisfy x2 − y2 = 210?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Organised cases
What is the smallest whole number n such that n! ends in exactly 100 zeros?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Extremal principle
What is the remainder when 12026 + 22026 + 32026 + … + 122026 is divided by 13?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Invariants, Proof techniques
What is the largest whole number that divides n5 − n for every whole number n?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Proof techniques, Extremal principle
How many triples of whole numbers (x, y, z), positive, negative or zero, satisfy x2 + y2 = 3z2?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Proof techniques, Extremal principle
How many positive divisors of 1010 are not divisors of 109?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Count the opposite
How many positive whole numbers n ≤ 1000 have the property that n divides 2n − 1? Prove your answer.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
What is the smallest positive whole number whose cube ends in the digits 111?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards, Organised cases
How many whole numbers x with 1 ≤ x ≤ 2026 have no common factor with 2026 (other than 1) and satisfy x2 ≡ 1 (mod 2026)? (1013 is prime.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
How many ordered pairs of whole numbers (x, y), positive, negative or zero, satisfy xy = 2x + 3y + 6?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Working backwards
What is the highest common factor of all numbers of the form p4 − 1, where p is a prime greater than 5?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Proof techniques, Extremal principle
How many ordered pairs (a, b) of positive whole numbers have lowest common multiple 23 × 32 × 5 = 360?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Count the opposite
Find the sum of all positive whole numbers n for which 1! + 2! + 3! + … + n! is a perfect square.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Organised cases
For how many whole numbers n with 1 ≤ n ≤ 2026 is ⌊n/2⌋ + ⌊n/3⌋ + ⌊n/6⌋ = n? (⌊y⌋ is the greatest whole number not more than y.)
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
Positive whole numbers a < b satisfy ab = ba. Find a + b, and show there is only one such pair.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Extremal principle, Proof techniques
For how many primes p does the decimal expansion of 1/p repeat with period exactly 6?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases, Proof techniques
How many whole numbers x with 0 ≤ x < 1000 satisfy x2 ≡ x (mod 1000)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Organised cases
How many whole numbers n with 1 ≤ n ≤ 2026 can be written as a sum of two or more consecutive positive whole numbers?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Proof techniques
Find the sum of all whole numbers n (positive, negative or zero) for which (n2 + 3n + 5)/(n − 1) is a whole number.
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Symmetry, Organised cases
What is the largest whole number k such that 7k divides the binomial coefficient C(2026, 1013)?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Spot the pattern and generalise, Proof techniques
For how many whole numbers n with 1 ≤ n ≤ 100 is 3n + 4n divisible by 7?
Hints, answer check and full worked solution. Included with every A Level, IB, IGCSE and CBSE plan.
Strategy: Parity and remainders, Spot the pattern and generalise
More Olympiad-style problems: Combinatorics · Geometry · Algebra · Probability · Logic · Calculus and functions
Number theory at other levels: Junior (ages 11 to 13) · Intermediate (ages 13 to 16) · Senior (ages 16 to 18)
Problem-solving strategies · Where next · Extension & competition maths
A Level Math Revision