My Math Forum 6 digit no.

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 January 7th, 2012, 10:42 AM #1 Senior Member   Joined: Jul 2011 Posts: 402 Thanks: 16 6 digit no. A $6$ Digit no. $abcdef$ is Multiplied by $6$. then the resulting no. is $defabc$ Then no. is
 January 7th, 2012, 01:38 PM #2 Senior Member     Joined: Feb 2010 Posts: 701 Thanks: 136 Re: 6 digit no. 142857
 January 7th, 2012, 09:14 PM #3 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs Re: 6 digit no. Consider the following: We might look for a repeating decimal x having six unique digits: $x=0.\bar{\text{abcdef}}$ $1000x=\text{abc}+0.\bar{\text{defabc}}$ We want $6x=0.\bar{\text{defabc}}$ $1000x=\text{abc}+6x$ $994x=\text{abc}$ $x=\frac{\text{abc}}{994}=\frac{\text{abc}}{2\cdot7 \cdot71}$ To get a repeating decimal of period 6, we need to let abc = 2·71 = 142, thus: $x=\frac{1}{7}$ and then: $\frac{1000}{7}=142+0.\bar{\text{def}142}$ $\frac{1000}{7}-142=0.\bar{\text{def}142}$ $\frac{6}{7}=0.\bar{857142}$ Thus, def = 857, and the number abcdef = 142857.
 January 8th, 2012, 02:09 AM #4 Math Team   Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,799 Thanks: 970 Re: 6 digit no. (100a + 10b + c) / (100d + 10e + f) = 994 / 5999
 January 8th, 2012, 09:07 AM #5 Senior Member   Joined: Jul 2011 Posts: 402 Thanks: 16 Re: 6 digit no. Thanks mrtwhs, Markfl and Denis
 January 10th, 2012, 02:11 PM #6 Newbie   Joined: Sep 2011 Posts: 29 Thanks: 0 Re: 6 digit no. It should be pointed out that this exact problem is in the (excellent book) Algebra by Israel M. Gelfand. The solution appears in said book, which is noticing that A must be 1 and working backwards from there. I have seen this problem elsewhere with no sources attributed to Gelfand. It is proper etiquette to cite sources when quoting from a popular algebra book. He has books for geometry, trigonometry, functions, as well.
 January 10th, 2012, 04:07 PM #7 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs Re: 6 digit no. This problem is probably in other books as well, and there is more than one way to solve it. To me, after some thought on it, a repeating decimal of period 6 seemed to be the way to go.
 January 10th, 2012, 07:43 PM #8 Newbie   Joined: Sep 2011 Posts: 29 Thanks: 0 Re: 6 digit no. I thought your way was really good and it reminds me of demonstrations of repeating rational numbers. I just thought if people found this in a search engine they might like to see this book, which has other similar problems. Also, to remember if you do cite a book written for a popular audience to remember to include the name.
 January 10th, 2012, 07:51 PM #9 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,209 Thanks: 517 Math Focus: Calculus/ODEs Re: 6 digit no. Oh, I wholeheartedly agree, if an interesting problem is found outside of a curricular textbook, the author and the title of the book should be cited, not just to give credit where it is due, but also so folks can check out the book for themselves if they are so inclined.

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