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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

Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

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.

More Sums of Squares

Stage: 5

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

Prime AP

Stage: 4 Challenge Level:

Show that if three prime numbers, all greater than 3, form an arithmetic progression then the common difference is divisible by 6. What if one of the terms is 3?

Modulus Arithmetic and a Solution to Differences

Stage: 5

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

What Numbers Can We Make Now?

Stage: 3 and 4 Challenge Level:

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

Modular Fractions

Stage: 5 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.

Mod 3

Stage: 4 Challenge Level:

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

Obviously?

Stage: 4 and 5 Challenge Level:

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

Old Nuts

Stage: 5 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?

Elevens

Stage: 5 Challenge Level:

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

Dirisibly Yours

Stage: 5 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.

Mod 7

Stage: 5 Challenge Level:

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

Latin Squares

Stage: 3, 4 and 5

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.

Weekly Challenge 41: Happy Birthday

Stage: 5 Challenge Level:

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

Days and Dates

Stage: 4 Challenge Level:

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

Knapsack

Stage: 4 Challenge Level:

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

Take Three from Five

Stage: 3 and 4 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?

Filling the Gaps

Stage: 4 Challenge Level:

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

The Public Key

Stage: 5 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.

Double Time

Stage: 5 Challenge Level:

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

Novemberish

Stage: 4 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.

Weekly Challenge 8: Sixinit

Stage: 5 Short Challenge Level:

Choose any whole number n, cube it and add 11n. Is the answer always divisible by 6? If so why?

More Mods

Stage: 4 Challenge Level:

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

Pythagoras Mod 5

Stage: 5 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.

Small Groups

Stage: 5

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

Stage: 5 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.

Grid Lockout

Stage: 4 Challenge Level:

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

Remainder Hunt

Stage: 5 Challenge Level:

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

Rational Round

Stage: 5 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.

Transposition Fix

Stage: 4 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. . . .

Check Code Sensitivity

Stage: 4 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.

Check Codes

Stage: 4 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. . . .

Zeller's Birthday

Stage: 4 Challenge Level:

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

Modular Knights

Stage: 5 Challenge Level:

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

Purr-fection

Stage: 5 Challenge Level:

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

Euler's Officers

Stage: 4 Challenge Level:

How many different solutions can you find to this problem? Arrange 25 officers, each having one of five different ranks a, b, c, d and e, and belonging to one of five different regiments p, q, r, s. . . .

The Knapsack Problem and Public Key Cryptography

Stage: 5

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.

Guesswork

Stage: 4 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.

Shuffles

Stage: 5 Challenge Level:

An environment for exploring the properties of small groups.

The Best Card Trick?

Stage: 3 and 4 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?