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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,930

Let m,n be positive integers. When m/n is expressed as a decimal, three consecutive digits after the decimal point are 1,6, 7 in order. Find the smallest possible value of n.

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You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**pi_cubed****Member**- From: A rhombicosidodecahedron
- Registered: 2020-06-22
- Posts: 63

Since 0.167=167/1000 and that is in the lowest terms, the smallest value of n must be 1000...

*Last edited by pi_cubed (2020-07-03 14:32:19)*

e to the i pi plus 1 is zero -Leonhard Euler | a squared plus b squared equals c squared -Pythagoras | Energy equals mass times the speed of light squared -Albert Einstein | i² = -1 | x^2+1=0 x^2=-1 x=i

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,930

hi pi_cubed

Remember this isn't my problem.

It says '3 digits are'; it's unclear if that means the first three or any three. It doesn't say the decimal is just 0.167 There may be more digits after that.

Assuming it's the first three, I have found an 'n' that is lower than 1000. I got 0.167989....

If the answer may be 0.dddd167dddd then I think I could get an even lower n.

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,930

401/491= 0.816700....

I am not yet claiming that is the lowest 'n'. Still working on it.

B

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,653

bob bundy wrote:

Assuming it's the first three, I have found an 'n' that is lower than 1000. I got 0.167989....Bob

Hi Bob,

For 167dddd..., I got m=22, n=131, m/n=0.**167**9389...

Did you omit the '3' that's in my quotient, or did you find a different *n* than 131?

Here's the complete list of lowest *n* I found:

167dddddd... : m=22, n=131, m/n=0.**167**938931...

d167ddddd... : m=107, n=131, m/n=0.8**167**93893...

dd167dddd... : m=50, n=131, m/n=0.38**167**9389...

ddd167ddd... : m=5, n=131, m/n=0.038**167**938...

dddd167dd... : m=66, n=131, m/n=0.5038**167**93...

ddddd167d... : m=59, n=131, m/n=0.45038**167**9...

dddddd167... : m=19, n=131, m/n=0.145038**167**...

And for all such {167dddddd,d167ddddd,...dddddd167}, I found that the next-to-lowest *n* is 137.

*Last edited by phrontister (2020-07-06 11:18:16)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 8,930

hi Phro,

Great to hear from you.

Just knocked together a short prog. to crunch this. Couldn't do better than 131.

Is there a way to get this analytically I wonder?

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,653

Hi Bob;

Nice to have another go at something.

I see that your last post came before I finished an edit to my previous post (#5), and so you might like to have a look at that one.

I wrote a small, clunky, BASIC program to get my results, and can't think of a cleverer approach.

It's interesting that all my results for the lowest *n* have n=131...which looks like a pattern to me.

But it all seems rather frivolous, and so I think I'll leave it there.

*Last edited by phrontister (2020-08-05 21:52:24)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**Raveeshjji****Member**- Registered: 2020-08-05
- Posts: 1

I am not yet claiming that is the lowest 'n'. Still working on it.

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