Resources tagged with: Number theory

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Broad Topics > Properties of Numbers > Number theory

An Introduction to Number Theory

Age 16 to 18

An introduction to some beautiful results in Number Theory.

Ordered Sums

Age 14 to 16Challenge 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. . . .

A Little Light Thinking

Age 14 to 16Challenge Level

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

Always Perfect

Age 14 to 18Challenge Level

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

Euler's Squares

Age 14 to 16Challenge Level

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

Data Chunks

Age 14 to 16Challenge 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. . . .

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.

Pythagorean Golden Means

Age 16 to 18Challenge 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.

Binomial Coefficients

Age 14 to 18

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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.

Really Mr. Bond

Age 14 to 16Challenge 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?

Never Prime

Age 14 to 16Challenge 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.

Filling the Gaps

Age 14 to 16Challenge Level

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

There's a Limit

Age 14 to 18Challenge 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?

Diophantine N-tuples

Age 14 to 16Challenge Level

Can you explain why a sequence of operations always gives you perfect squares?

Mod 7

Age 16 to 18Challenge Level

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

More Sums of Squares

Age 16 to 18

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

Number Rules - OK

Age 14 to 16Challenge Level

Can you produce convincing arguments that a selection of statements about numbers are true?

Novemberish

Age 14 to 16Challenge 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.

An Introduction to Modular Arithmetic

Age 14 to 18

An introduction to the notation and uses of modular arithmetic

The Public Key

Age 16 to 18Challenge 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.