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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.)

  Submitted by e.g.    
Rating: 3.3333 (3 votes)
Solution: (Hide)
I'll show this as an algorithm.
Start with N=2^76 (given)
Set K=N
Set I=1
While I isn't greater that 76
    look at the I-th digit of N, from the right
    while it isn't a 6 or 7, add K to N
    Multiply K by 10
    Add 1 to I
Take the 76 leftmost digits of N as the answer.
At the end (before taking the 76 leftmost digits), N can be written as Px10^76+Q, and Q is written with sixes and sevens only. Since N and Px10^76 are multiple of 2^76, it follows that Q also is a multiple of 2^76, and that's the why of the last instruction of the algorithm.

On average, the "add K to N" order is done twice, so we need about 152 sums to solve the problem.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionMy solutionFederico Kereki2004-07-17 21:36:32
Some Thoughtsre: No SubjectFederico Kereki2004-07-16 18:32:00
No SubjectCory Taylor2004-07-16 17:23:59
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