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Odd Differences

The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.


How many zeros are there at the end of the number which is the product of first hundred positive integers?

Rachel's Problem

Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!

Triangular Triples

Age 14 to 16 Challenge Level:

Three numbers a, b and c are a Pythagorean triple if $a^2+ b^2= c^2$. The triangular numbers are:

$\frac{1\times 2}{2}, \frac{2\times 3}{2}, \frac{3\times 4}{2}, \frac{4\times 5}{2}$

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

[In fact, these are the ONLY known set of three triangular numbers that form a Pythagorean triple.]