Oh! Hidden Inside?
Problem
Find the number which has 8 factors, such that the product of all the factors is 331776.
Remember: A a factor (or divisor) is a number that divides exactly into another number.
Student Solutions
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.
| x | 1 | a | aa | aaa |
| 1 | 1 | a | aa | aaa |
| b | b | ab | aab | aaab |
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