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.

Harmonic Triangle

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

Weekly Problem 26 - 2008

Stage: 3 and 4 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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Working systematically. Mathematical reasoning & proof. chemistry. Polynomials. Algebra - generally. Generalising. Simultaneous equations. Manipulating algebraic expressions/formulae. Algebraic fractions. Factorisation (algebraic).

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