Egyptian Fractions

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Fractions of Fractions

Weekly Problem 44 - 2013
If you know that a fraction of X is the same as a different fraction of Y, can you work out X/Y?

Harmonic Triangle

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Leftovers

Stage: 3 and 4 Short Challenge Level:

Note that

${n^2 \over n+4}={n^2+16 \over n+4}-{16 \over n+4}={(n+4)(n-4) \over n+4}-{16 \over n+4}=n-4-{16 \over n+4}$.

So when $n> 12$, the remainder when $n^2$ is divided by $n+4$ is always $16$.

For $1 \leq n \leq 12$, the remainder when $n^2$ is divided by $n+4$ is shown in the table below.

 $n\quad\quad\quad$ 1 2 2 4 5 6 7 8 9 10 11 12 $n+4$ 5 6 7 8 9 10 11 12 13 14 15 16 remainder 1 4 2 0 7 6 5 4 3 2 1 0

So there are $9$ different remainders, namely $0, 1, 2, 3, 4, 5, 6, 7, 16$.

This problem is taken from the UKMT Mathematical Challenges.
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