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Resources tagged with Modulus arithmetic similar to Modulus Arithmetic and a Solution to Dirisibly Yours:

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Broad Topics > Numbers and the Number System > Modulus arithmetic

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Modulus Arithmetic and a Solution to Dirisibly Yours

Age 16 to 18

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.

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Prime AP

Age 16 to 18 Challenge Level:

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

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More Sums of Squares

Age 16 to 18

Tom writes about expressing numbers as the sums of three squares.

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Modulus Arithmetic and a Solution to Differences

Age 16 to 18

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

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Modular Fractions

Age 16 to 18 Challenge Level:

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.

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Mod 3

Age 14 to 16 Challenge Level:

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

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Obviously?

Age 14 to 18 Challenge Level:

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

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Mod 7

Age 16 to 18 Challenge Level:

Find the remainder when 3^{2001} is divided by 7.

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Old Nuts

Age 16 to 18 Challenge Level:

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?

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The Public Key

Age 16 to 18 Challenge Level:

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.

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More Mods

Age 14 to 16 Challenge Level:

What is the units digit for the number 123^(456) ?

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Dirisibly Yours

Age 16 to 18 Challenge Level:

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.

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Elevens

Age 16 to 18 Challenge Level:

Add powers of 3 and powers of 7 and get multiples of 11.

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Remainder Hunt

Age 16 to 18 Challenge Level:

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

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Purr-fection

Age 16 to 18 Challenge Level:

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

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Latin Squares

Age 11 to 18

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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Filling the Gaps

Age 14 to 16 Challenge Level:

Which numbers can we write as a sum of square numbers?

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Knapsack

Age 14 to 16 Challenge Level:

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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Zeller's Birthday

Age 14 to 16 Challenge Level:

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

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Double Time

Age 16 to 18 Challenge Level:

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

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Grid Lockout

Age 14 to 16 Challenge Level:

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

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Sixinit

Age 16 to 18 Short Challenge Level:

Choose any whole number n, cube it, add 11n, and divide by 6. What do you notice?

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Readme

Age 16 to 18 Challenge Level:

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.

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Check Codes

Age 14 to 16 Challenge Level:

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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Rational Round

Age 16 to 18 Challenge Level:

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

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Small Groups

Age 16 to 18

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

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Check Code Sensitivity

Age 14 to 16 Challenge Level:

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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Transposition Fix

Age 14 to 16 Challenge Level:

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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Pythagoras Mod 5

Age 16 to 18 Challenge Level:

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.

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Novemberish

Age 14 to 16 Challenge Level:

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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Take Three from Five

Age 14 to 16 Challenge Level:

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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Weekly Challenge 41: Happy Birthday

Age 16 to 18 Challenge Level:

A weekly challenge concerning the interpretation of an algorithm to determine the day on which you were born.

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Euler's Officers

Age 14 to 16 Challenge Level:

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

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Guesswork

Age 14 to 16 Challenge Level:

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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Modular Knights

Age 16 to 18 Challenge Level:

Try to move the knight to visit each square once and return to the starting point on this unusual chessboard.

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The Knapsack Problem and Public Key Cryptography

Age 16 to 18

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.

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The Best Card Trick?

Age 11 to 16 Challenge Level:

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?

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Shuffles

Age 16 to 18 Challenge Level:

An environment for exploring the properties of small groups.