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Tenacious Divisibility Treat (Posted on 2016-04-06) Difficulty: 2 of 5
A positive integer less than 30 million is such that if we subtract 5 from it – the resulting number is divisible by 8.

At the first step, from the number (considered originally) diminished by 5 - we subtract the eighth part. We then obtain a number that also becomes divisible by 8 after 5 is subtracted from it.

At the second step, we derive another in the same way, namely by subtracting a eighth part from the number at the end of the first step diminished by 5. The resulting number is also divisible by 8 after subtracting 5.

The operation concludes at 8th step given that at the end of 7th step we get a number that is divisible by 8 after after subtracting 5.

Determine the positive integer initially before the first step.

*** The resulting number at the end of 8th step is NOT necessarily divisible by 8 after subtracting 5.

No Solution Yet Submitted by K Sengupta    
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Solution re: Help from a negative (solution) | Comment 6 of 9 |
(In reply to Help from a negative (solution) by Brian Smith)

I agree.

In longhand: x=8(8(8(8(8(8(8(8/7*k+5)/7+5)/7+5)/7+5)/7+5)/7+5)/7+5)/7+5, for some {x,k}.

Then 5764801x = (16777216k+385434525), when k = 7(823543n+823538),   x = 16777216n+16777181, with minimum at 16777181. Similarly, the last amount deducted is 823538, implying that the final remainder is seven times that, or 5764766.

Edited on April 7, 2016, 6:15 am
  Posted by broll on 2016-04-07 06:14:49

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