Diophantine equations

There are 25 NRICH Mathematical resources connected to Diophantine equations
CD Heaven
problem
Favourite

CD Heaven

Age
14 to 16
Challenge level
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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 each price?

Letter Land
problem
Favourite

Letter land

Age
11 to 14
Challenge level
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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.
Fibs
problem
Favourite

Fibs

Age
11 to 14
Challenge level
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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?
Whole Numbers Only
problem

Whole numbers only

Age
11 to 14
Challenge level
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Can you work out how many of each kind of pencil this student bought?
In particular
problem

In particular

Age
14 to 16
Challenge level
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Can you find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers?
Plutarch's Boxes
problem

Plutarch's boxes

Age
11 to 14
Challenge level
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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 their surface areas equal to their volumes?
Shades of Fermat's Last Theorem
problem

Shades of Fermat's Last Theorem

Age
16 to 18
Challenge level
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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?

Deep Roots
problem

Deep roots

Age
14 to 16
Challenge level
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Find integer solutions to: $\sqrt{a+b\sqrt{x}} + \sqrt{c+d.\sqrt{x}}=1$
Upsetting Pitagoras
problem

Upsetting Pythagoras

Age
14 to 18
Challenge level
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Find the smallest integer solution to the equation 1/x^2 + 1/y^2 = 1/z^2

Rudolff's Problem
problem

Rudolff's problem

Age
14 to 16
Challenge level
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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?