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Upsetting Pitagoras

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

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Euclid's Algorithm I

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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What's a Group?

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.

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Age 16 to 18 Challenge Level:

Why do this problem?

The problem gives practice in working with linear inequalities and in working systematically through separate cases.

Possible approach

Encourage the class to try some numerical values for $x$, to compare the values of the three functions and to record their findings. Collect sufficient results from the class to provide evidence for spotting patterns and making conjectures.

Key questions

If the integer part of $x$ is $a$ then $x=a + b$ where $a$ is a whole number and $0\leq b < 1$. What is the difference between the separate cases where $0 \leq b < {1\over 2}$ and ${1\over 2}\leq b < 1$?