Enigma 1703: G&Ts all round
20 June 2012
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From New Scientist #2870, 23rd June 2012 [link]
I was staying at my sister’s house when my niece Amy came home from school feeling special. The class had been shown how to split a whole number, T, into whole number parts in such a way that the product of the parts was the greatest, G, that could be obtained for that T. For instance, she explained, 10 could be split into ten ones, or 2 and 4 and 4, or 5 and 5, and so on, which would yield products of 1, 32, and 25 respectively. But, she warned, G exceeds 32 for T=10.
Why did she feel special? Well, each pupil in the class had been given a different number in the range 20-50 inclusive for their personal T, and she had noted that, when she added the digits of her G together, the sum was exactly half of her T, and no one else in the class had T and G with this property.
What value of T was Amy given?