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Broad Topics > Algebra > Diophantine equations

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Letter Land

Stage: 3 Challenge Level: Challenge Level:1

If: A + C = A; F x D = F; B - G = G; A + H = E; B / H = G; E - G = F and A-H represent the numbers from 0 to 7 Find the values of A, B, C, D, E, F and H.

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CD Heaven

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

All CD Heaven stores were given the same number of a popular CD to sell for £24. In their two week sale each store reduces the price of the CD by 25% ... How many CDs did the store sell at. . . .

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Fibs

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

The well known Fibonacci sequence is 1 ,1, 2, 3, 5, 8, 13, 21.... How many Fibonacci sequences can you find containing the number 196 as one of the terms?

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Solving with Euclid's Algorithm

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

A java applet that takes you through the steps needed to solve a Diophantine equation of the form Px+Qy=1 using Euclid's algorithm.

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Euclid's Algorithm II

Stage: 5

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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Euclid's Algorithm I

Stage: 5

How can we solve equations like 13x + 29y = 42 or 2x +4y = 13 with the solutions x and y being integers? Read this article to find out.

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Why Stop at Three by One

Stage: 5

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

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Are You Kidding

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

If the altitude of an isosceles triangle is 8 units and the perimeter of the triangle is 32 units.... What is the area of the triangle?

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Deep Roots

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$

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Plutarch's Boxes

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have. . . .

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In Particular

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Then find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers.

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Whole Numbers Only

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you work out how many of each kind of pencil this student bought?

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Not a Polite Question

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

When asked how old she was, the teacher replied: My age in years is not prime but odd and when reversed and added to my age you have a perfect square...

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Coffee

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

To make 11 kilograms of this blend of coffee costs £15 per kilogram. The blend uses more Brazilian, Kenyan and Mocha coffee... How many kilograms of each type of coffee are used?

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Hallway Borders

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

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Double Angle Triples

Stage: 5 Challenge Level: Challenge Level:1

Try out this geometry problem involving trigonometry and number theory

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Our Ages

Stage: 4 Challenge Level: Challenge Level:1

I am exactly n times my daughter's age. In m years I shall be exactly (n-1) times her age. In m2 years I shall be exactly (n-2) times her age. After that I shall never again be an exact multiple of. . . .

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Code to Zero

Stage: 5 Challenge Level: Challenge Level:1

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

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Exhaustion

Stage: 5 Challenge Level: Challenge Level:1

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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BT.. Eat Your Heart Out

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

If the last four digits of my phone number are placed in front of the remaining three you get one more than twice my number! What is it?

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Rudolff's Problem

Stage: 4 Challenge Level: Challenge Level:1

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

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Upsetting Pitagoras

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

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Shades of Fermat's Last Theorem

Stage: 5 Challenge Level: Challenge Level:1

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?