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 Tenacious Divisibility Treat (Posted on 2016-04-06)
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 No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
 broll's interpretation is correct | Comment 8 of 9 |
Starting with broll's 16777181

st = 16777181
For rpt = 1 To 8
newone = (st - 5) * 7 / 8
Text1.Text = Text1.Text & newone & crlf
st = newone
Next

we do get a valid sequence:

14680029 ... after first operation
12845021
11239389
9834461
8605149
7529501
6588309
5764766 ... after 8th operation

From  11,983,725

10485755
9175031.25
8028147.96875
7024625.09765625
6146542.58544922
5378220.38726807
4705938.46385956
4117691.78087711

From  28,760,941

25165819
22020087.25
19267571.96875
16859121.0976563
14751726.5854492
12907756.3872681
11294282.4638596
9882492.78087711

all with non-integral values.

I believe Ady made the same mistake I made: dividing by 8 rather than multiplying by 7/8 at each step after subtracting 5.  I retracted that wrong solution. If one made that wrong choice, the one would indeed get

1497965
187245
23405
2925
365
45
5
0

or

3595117
449389
56173
7021
877
109
13
1

respectively.

 Posted by Charlie on 2016-04-07 10:41:07

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