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Topic: Number with prime factors 3 & 7, ending in 11 (Read 710 times) |
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gkwal
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Number with prime factors 3 & 7, ending in 11
« on: May 3rd, 2007, 11:35am » |
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Show that no positive integer exists whose prime factors are at most 3 and 7, and which ends in the digits 11.
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Grimbal
wu::riddles Moderator
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Re: Number with prime factors 3 & 7, ending in
« Reply #1 on: May 3rd, 2007, 1:42pm » |
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What we are looking is a
N = 2a·3b·5c·7d
That ends in 11.
To end in 1, a and c must be zero.
=> N = 3b·7d
Let's consider it modulo 20. We want to find such an N that equals 11 (mod 20).
33 = 27 = 7 (mod 20)
so 7d = 33d
=> N = 3b+3d (mod 20)
But the powers of 3 (mod 20) are 1, 3 ,9, 7, 1, ...
so there is no way to get 11.
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