Other tags that relate to Function Pyramids
Graph sketching. Mathematical reasoning & proof. Powers & roots. Working systematically. Surds. Functions and their inverses. Modular arithmetic. Biology. Exponential and Logarithmic Functions. Chemistry.There are 39 results
Broad Topics > Numbers and the Number System > Modular arithmetic
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.
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.
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?
The Chinese Remainder Theorem
Age 14 to 18
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."
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.
Filling the Gaps
Age 14 to 16 Challenge Level:
Which numbers can we write as a sum of square numbers?
An Introduction to Modular Arithmetic
Age 14 to 16
An introduction to the notation and uses of modular arithmetic
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?
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.
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.
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.
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.
Euler's Officers
Age 14 to 16 Challenge Level:
How many different ways can you arrange the officers in a square?
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.
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.
Purr-fection
Age 16 to 18 Challenge Level:
What is the smallest perfect square that ends with the four digits 9009?
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.
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?
Odd Stones
Age 14 to 16 Challenge Level:
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.
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.
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.
Grid Lockout
Age 14 to 16 Challenge Level:
What remainders do you get when square numbers are divided by 4?
Happy Birthday
Age 16 to 18 Challenge Level:
Can you interpret this algorithm to determine the day on which you were born?
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.
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?
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. . . .
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.
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.
Prime AP
Age 16 to 18 Challenge Level:
What can you say about the common difference of an AP where every term is prime?
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.
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.
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. . . .
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.
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.
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?
NRICH is part of the family of activities in the Millennium Mathematics Project.