There are **30** NRICH Mathematical resources connected to **Modular arithmetic**, you may find related items under Properties of Numbers.

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

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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?

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

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

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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.

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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.

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

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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?

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

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

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

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

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

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."

An introduction to the notation and uses of modular arithmetic

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

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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. . . .

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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.

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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. . . .

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

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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?

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.

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

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

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

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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.

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