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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