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Spirit of '76 (Posted on 2004-03-14) Difficulty: 3 of 5
Can you find a 76-digit multiple of 276, written exclusively with 7s and 6s?

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

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Solution Answer | Comment 4 of 8 |

6667776766677767667676666776766667777767666677766776777777777777666766667776 = 88247290723637794367680048552939376696792027490086866 × 276.

In general, there exists a k-digit number, written exclusively with 7s and 6s, divisible by 2k.  The proof is by induction.

For k = 1, 6 is divisible by 21.

Let nk be a k-digit number, written exclusively with 7s and 6s, divisible by 2k.

If nk = 0 (mod 2k), then nk = 0 or 2k (mod 2k+1).

Note that:
6×10k = 0 (mod 2k+1)
7×10k = 2k (mod 2k+1)

So we can choose, respectively, nk+1 = 6×10k + nk or 7×10k + nk = 0 (mod 2k+1).

Hence result.

The induction gives an explicit means of constructing nk.

Edited on March 14, 2004, 2:09 pm
  Posted by Nick Hobson on 2004-03-14 13:48:27

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