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00 Separate (Posted on 2003-10-10) Difficulty: 1 of 5
Prove that the numbers that begin with 1 and end with 1, with any number of 2s in the middle, and all 1s and 2s separated by 00 are composite.

For example:
1002001
1002002001,
1002002002001,
1002002002002001

See The Solution Submitted by Gamer    
Rating: 3.0000 (7 votes)

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A solution | Comment 3 of 13 |
The first term is a multiple of 1001 (1002001 = 1001*1001). The nth term of the sequence can be expressed as a recursion: t(n) = 1000*t(n-1)+1001. t(n) is a multiple of 1001 if and only if t(n-1) also is a multiple of 1001. Since t(1) is a multiple of 1001 then t(2) is a multiple of 1001. Since t(2) is a multiple of 1001 then t(3) is a multiple of 1001. Etc...
  Posted by Brian Smith on 2003-10-10 15:22:26
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