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Resources tagged with Modulus arithmetic similar to Be Reasonable:

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Broad Topics > Numbers and the Number System > Modulus arithmetic

Prime AP

Stage: 5 Challenge Level:

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

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.

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.

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.

More Sums of Squares

Stage: 5

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

Filling the Gaps

Stage: 4 Challenge Level:

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

Take Three from Five

Stage: 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?

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.

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.

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 7

Stage: 5 Challenge Level:

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

Elevens

Stage: 5 Challenge Level:

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

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.

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.

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.

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.

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?

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.

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?

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.

Purr-fection

Stage: 5 Challenge Level:

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

Sixinit

Stage: 5 Short Challenge Level:

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

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.

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.

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. . . .

Remainder Hunt

Stage: 5 Challenge Level:

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

Grid Lockout

Stage: 4 Challenge Level:

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

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. . . .

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.

Euler's Officers

Stage: 4 Challenge Level:

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

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.

Small Groups

Stage: 5

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

More Mods

Stage: 4 Challenge Level:

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

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.

Shuffles

Stage: 5 Challenge Level:

An environment for exploring the properties of small groups.

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.

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.