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Niners (Posted on 2002-10-23) Difficulty: 4 of 5
The set of numbers {9, 99, 999, 9999, ...} has some interesting properties. One of these has to do with factorization. Take any number n that isn't divisible by 2 or by 5. You will be able to find at least one number in the set that is divisible by n. Furthermore, you won't need to look beyond the first n numbers in the set.

Prove it.

(from http://www.ocf.berkeley.edu/~wwu/riddles/)

See The Solution Submitted by levik    
Rating: 4.2500 (8 votes)

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re(2): | Comment 5 of 15 |
(In reply to re: by levik)

that is a good point, which arises because i didnt
explain this well:

"because the property of differences, each N()
must have a different r."

consider that if two elements N(n), N(n')
in N(1)..N(p) have the same r, it follows that the
difference between them is exactly divisible by p.

N(n) % p = r and N(n') % p = r -> N(n) - N(n') % p = 0

because of the differences property:

10^x * N(n-n') % p = 0. if p is neither 2m or 5m, then
10^x * N(n-n') % p = 0 -> N(n-n') % p = 0,
and
N(n-n') < N(p)

in words, if two of the remainders for N(1)..N(p) were
the same, then their difference would be divisible. but
because the difference is itself a member of N, then
there would be some N() divisible by p.
this argument breaks down for p = 2m or 5m because of
the 10^x term in

N(n) - N(n') = 10^x * N(n-n')

which contains 2,5 factors


  Posted by Cheradenine on 2002-10-23 22:56:37

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