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Weekly Problem 26 - 2008
If $n$ is a positive integer, how many different values for the remainder are obtained when $n^2$ is divided by $n+4$?

Harmonic Triangle

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

$$\frac{1}{2} = \frac{1}{3} + \frac{1}{6}$$
$$\frac{1}{3} = \frac{1}{4} + \frac{1}{12}$$
$$\frac{1}{12} = \frac{1}{20} + \frac{1}{30}$$

Look at the diagonal lines running from the right down to the left. The fractions in the first one are $\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and so on.

If $\frac{1}{n}$ is at the end of the nth row, the fraction above it must be $\frac{1}{n-1}$ and the fraction below it must be $\frac{1}{n+1}$.

Have a look at the second diagonal (the one formed by taking the second number in each row: it starts $\frac{1}{2}, \frac{1}{6}, \frac{1}{12}, \frac{1}{20}$.

Can you find a pattern for these numbers so that you can work them out easily (without having to subtract fractions)?

Can you explain why the pattern works?