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38s and Bunches of 1s (Posted on 2017-06-30) Difficulty: 3 of 5
Prove every number in the sequence 38, 381, 3811, 38111, 381111, ... is composite.

See The Solution Submitted by Brian Smith    
Rating: 4.5000 (2 votes)

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Solution Proof Comment 7 of 7 |
If the number of 1's is 3*n+1, then the digits add up to 3*n+12. Therefore, the number is divisible by 3. If the number of 1's is 3*n+2, then the number is divisible by 37. Suppose the number of 1's is 3*n. Then, the number is ((343*(10^(3*n)))-1)/9=(((7*(10^n))-1)/3)*(((49*(10^(2*n)))+(7*(10^n))+1)/3). Therefore, the number is always composite.

  Posted by Math Man on 2017-07-01 21:38:12
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