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## 'Oh! Hidden Inside?' printed from http://nrich.maths.org/

The solution is based on the divisors of 24.

i.e 1 x 2 x 3 x 4 x 6 x 8 x 12 x 24 = 331776

*Alison
at Maidstone Girls Grammar School arrived at her
solution* by trial and
improvement techniques and had found that the divisors of 30 were
too big when multiplied together and the divisors of 20 too small.
She had ascertained that 24 did have quite a lot of divisors
compared to others in this range.

*Serendipity helped Ashley and John from the
Simon Langton Grammar School who "luckily chose the right number
(24)" .* But, when
using prime factors, they found that 24 = 2 x 2 x 2 x 3 and they
noticed that numbers whose prime factors are of the form a x a x a
x b have ONLY 8 divisors as they showed in a multiplication table,
reproduced below:

x |
1 |
a |
aa |
aaa |

1 |
1 |
a |
aa |
aaa |

b |
b |
ab |
aab |
aaab |

To find the answer is then just a
matter of substituting different prime numbers for a and b
.

Similarly Robert (Smithdon High School) had
relied upon the prime factorisation of 331776 for his
solution.

*Gareth
rom Hethersett High School, Norwich used algebra and found the
fourth root of 331776, namely 24 - sort of hidden
inside:*

Let a, b, c, d, e, f, g, and h be the divisors where a x b x c x
d x e x f x g x h = 331776 and a = 1.

But a x h = h; b x g = h; c x f = h and d x e = h

hence h x h x h x h = 331776

and so h = 24