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There are **34** NRICH Mathematical resources connected to **Number theory**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### Overlaps

Can you find ways to put numbers in the overlaps so the rings have equal totals?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Impossibilities

Just because a problem is impossible doesn't mean it's difficult...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Little Light Thinking

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Much Can We Spend?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Always Perfect

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Number Rules - OK

Can you produce convincing arguments that a selection of statements about numbers are true?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Where Can We Visit?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Differences

Can you guarantee that, for any three numbers you choose, the product of their differences will always be an even number?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Age 14 to 18

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Binomial Coefficients

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

Age 14 to 18

Article
Primary curriculum
Secondary curriculum
### An Introduction to Number Theory

An introduction to some beautiful results in Number Theory.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### An Introduction to Modular Arithmetic

An introduction to the notation and uses of modular arithmetic

Age 14 to 18

Problem
Primary curriculum
Secondary curriculum
### Data Chunks

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and blue chunks, explore what sizes near to 31 can, or cannot, be exactly filled.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Got It Article

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

Age 7 to 14

Problem
Primary curriculum
Secondary curriculum
### The Public Key

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A One in Seven Chance

What is the remainder when 2^{164}is divided by 7?

Age 11 to 14

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Powerful Properties

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### About Pythagorean Golden Means

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio?

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### The Codabar Check

This article explains how credit card numbers are defined and the check digit serves to verify their accuracy.

Age 11 to 14

Article
Primary curriculum
Secondary curriculum
### Sums of Squares and Sums of Cubes

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

Age 16 to 18

Article
Primary curriculum
Secondary curriculum
### Modulus Arithmetic and a Solution to Differences

Peter Zimmerman, a Year 13 student at Mill Hill County High School in Barnet, London wrote this account of modulus arithmetic.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Really Mr. Bond

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Strange Numbers

All strange numbers are prime. Every one digit prime number is strange and a number of two or more digits is strange if and only if so are the two numbers obtained from it by omitting either its first or its last digit. Find all strange numbers.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Never Prime

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Marbles

I start with a red, a blue, a green and a yellow marble. I can trade any of my marbles for three others, one of each colour. Can I end up with exactly two marbles of each colour?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Marbles

I start with a red, a green and a blue marble. I can trade any of my marbles for two others, one of each colour. Can I end up with five more blue marbles than red after a number of such trades?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Diophantine N-tuples

Can you explain why a sequence of operations always gives you perfect squares?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagorean Golden Means

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Euler's Squares

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...

Age 14 to 16

Challenge Level