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There are **51** NRICH Mathematical resources connected to **Modular arithmetic**, you may find related items under Properties of numbers.

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make Now?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### What Numbers Can We Make?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Going Round in Circles

Mathematicians are always looking for efficient methods for solving problems. How efficient can you be?

Age 11 to 14

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
### Elevenses

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Round and Round and Round

Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 รท 360. How did this help?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Odd Stones

On a "move" a stone is removed from two of the circles and placed in the third circle. Here are five of the ways that 27 stones could be distributed.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Guesswork

Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take Three from Five

Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

Age 11 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
### Prime AP

What can you say about the common difference of an AP where every term is prime?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Happy birthDay

Can you interpret this algorithm to determine the day on which you were born?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Zeller's Birthday

What day of the week were you born on? Do you know? Here's a way to find out.

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### The Chinese Remainder Theorem

In this article we shall consider how to solve problems such as "Find all integers that leave a remainder of 1 when divided by 2, 3, and 5."

Age 14 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

Article
Primary curriculum
Secondary curriculum
### The Knapsack Problem and Public Key Cryptography

An example of a simple Public Key code, called the Knapsack Code is described in this article, alongside some information on its origins. A knowledge of modular arithmetic is useful.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Knapsack

You have worked out a secret code with a friend. Every letter in the alphabet can be represented by a binary value.

Age 14 to 16

Challenge Level

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
### Double Time

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Readme

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Modular Fractions

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Transposition Fix

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Check Code Sensitivity

You are given the method used for assigning certain check codes and you have to find out if an error in a single digit can be identified.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Check Codes

Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Obviously?

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Age 14 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Pythagoras Mod 5

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Rational Round

Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dirisibly Yours

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Age 16 to 18

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

Problem
Primary curriculum
Secondary curriculum
### The Best Card Trick?

Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?

Age 11 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Latin Squares

A Latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and exactly once in each column.

Age 11 to 18

Article
Primary curriculum
Secondary curriculum
### Modulus Arithmetic and a Solution to Dirisibly Yours

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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

Article
Primary curriculum
Secondary curriculum
### Small Groups

Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.

Age 16 to 18

Problem
Primary curriculum
Secondary curriculum
### Two Much

Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Grid Lockout

What remainders do you get when square numbers are divided by 4?

Age 14 to 16

Challenge Level

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

How many different ways can you arrange the officers in a square?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Remainder Hunt

What are the possible remainders when the 100-th power of an integer is divided by 125?

Age 16 to 18

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
### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Purr-fection

What is the smallest perfect square that ends with the four digits 9009?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### It Must Be 2000

Here are many ideas for you to investigate - all linked with the number 2000.

Age 7 to 11

Challenge Level