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Topic: Years young (Read 773 times) |
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pcbouhid
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Years young
« on: Jan 10th, 2006, 8:27am » |
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Paul recently had another birthday. When someone mentioned that he was getting up there in years, he replied that he was actually quite young.
Indeed, he pointed out, he is the youngest age such that the sum of the divisors of his age, not including the age itself, exceeded his age, yet the sum of no subset of these divisors equaled his age.
How old had Paul just turned?
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Sir Col
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Re: Years young
« Reply #1 on: Jan 11th, 2006, 11:14am » |
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(Nice puzzle!)
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We are looking for abundant numbers: sum of proper divisors exceed number, which up to a maximum sensible age of around 120 is quite extensive.
12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ...
However, we can reduce the search considerably. It cannot be a multiple of 6, as n/6 + n/3 + n/2 = n; similarly it cannot be a multiple of 20, as n/20 + n/5 + n/4 + n/2 = n.
This leaves: 56, 70, 88, 104, ...
A quick check shows that 56 = 28 + 14 + 8 + 4 + 2, but with 70: 1, 2, 5, 7, 10, 14, 35, there is no such subset sum.
Hence Paul must have just turned 70 years of age.
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pcbouhid
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Re: Years young
« Reply #2 on: Jan 11th, 2006, 11:25am » |
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Right, Sir Col. These numbers are called weird numbers by the mathematicians, and the next smallest are 836, 4030, 5830 and 7192.
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Noke Lieu
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Re: Years young
« Reply #3 on: Jan 11th, 2006, 3:06pm » |
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at last. A way of being weird without being odd.
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