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Spirit of 76 with no products (Posted on 2004-07-13) Difficulty: 3 of 5
Can you find a 76 digits multiple of 2^76, written exclusively with sixes and sevens (as asked in "Spirit of 76"), starting with the (given) value of 2^76, and doing nothing but a (relatively) few sums?

(To be more precise: you should be able to do this by hand, with just pencil and paper, in a reasonably short time -- minutes, not hours or days.)

See The Solution Submitted by e.g.    
Rating: 3.3333 (3 votes)

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Solution My solution Comment 3 of 3 |
My algorithm:

Start with N=0, and S=2^76 (given).
Do the following 76 times, for I=1 to 76
while the I-th digit of N (from the right)
isn't 6 or 7, keep adding S to N
multiply S by 10

After the 76 times, the last (right) 76 digits of N are the solution

To see why this works, obviously N is always a multiple of 2^76, and taking the right 76 digits is ths same as taking N modulus 10^76 -- but as 10^76 is a multiple of 2^76, the modulus is a multiple of 2^76 too.

  Posted by Federico Kereki on 2004-07-17 21:36:32
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