# Resources tagged with: Number theory

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Broad Topics > Numbers and the Number System > Number theory ### 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. ### More Sums of Squares

##### Age 16 to 18

Tom writes about expressing numbers as the sums of three squares. ### Mod 7

##### Age 16 to 18 Challenge Level:

Find the remainder when 3^{2001} is divided by 7. ### Pythagorean Golden Means

##### Age 16 to 18 Challenge Level:

Show that the arithmetic mean, geometric mean and harmonic mean of a and b can be the lengths of the sides of a right-angles triangle if and only if a = bx^3, where x is the Golden Ratio. ### An Introduction to Number Theory

##### Age 16 to 18

An introduction to some beautiful results of Number Theory (a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions) ### Sums of Squares and Sums of Cubes

##### Age 16 to 18

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes. ### 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. ### 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. ### 2^n -n Numbers

##### Age 16 to 18

Yatir from Israel wrote this article on numbers that can be written as $2^n-n$ where n is a positive integer. ### An Introduction to Modular Arithmetic

##### Age 14 to 16

An introduction to the notation and uses of modular arithmetic ### Number Rules - OK

##### Age 14 to 16 Challenge Level:

Can you convince me of each of the following: If a square number is multiplied by a square number the product is ALWAYS a square number... ### Binomial Coefficients

##### Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies. ### There's a Limit

##### Age 14 to 18 Challenge Level:

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely? ### Really Mr. Bond

##### Age 14 to 16 Challenge Level:

115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise? ### Always Perfect

##### Age 14 to 16 Challenge Level:

Show that if you add 1 to the product of four consecutive numbers the answer is ALWAYS a perfect square. ### Ordered Sums

##### Age 14 to 16 Challenge Level:

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate. . . . ##### Age 16 to 18

What is the relationship between the arithmetic, geometric and harmonic means of two numbers, the sides of a right angled triangle and the Golden Ratio? ### Never Prime

##### Age 14 to 16 Challenge Level:

If a two digit number has its digits reversed and the smaller of the two numbers is subtracted from the larger, prove the difference can never be prime. ### Data Chunks

##### Age 14 to 16 Challenge Level:

Data is sent in chunks of two different sizes - a yellow chunk has 5 characters and a blue chunk has 9 characters. A data slot of size 31 cannot be exactly filled with a combination of yellow and. . . . ### Diophantine N-tuples

##### Age 14 to 16 Challenge Level:

Can you explain why a sequence of operations always gives you perfect squares? ### A Little Light Thinking

##### Age 14 to 16 Challenge Level:

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights? ### Euler's Squares

##### Age 14 to 16 Challenge Level:

Euler found four whole numbers such that the sum of any two of the numbers is a perfect square...