Given two positive integer numbers m,n, show that m/n can't have period b−1 in base b.
Given two positive integer numbers $m,n$, show that $frac{m}{n}$ can't have period $b-1$ in base $b$.
MY ATTEMPT: wlog $m<n$, then suppose $frac{m}{n}=0,a_1a_2...a_{b-1}$. Multiplying both sides by $b^{b-1}-1$ we obtain $frac{m(b^{b-1}-1)}{n}=a_1...a_{b-1}$ with $a_i in {0,1,2,...,b-1}$. How can I conclude?
number-theory
asked Jan 4 at 13:14
LanceLance
8912
add a comment |
Given two positive integer numbers $m,n$, show that $frac{m}{n}$ can't have period $b-1$ in base $b$.
MY ATTEMPT: wlog $m<n$, then suppose $frac{m}{n}=0,a_1a_2...a_{b-1}$. Multiplying both sides by $b^{b-1}-1$ we obtain $frac{m(b^{b-1}-1)}{n}=a_1...a_{b-1}$ with $a_i in {0,1,2,...,b-1}$. How can I conclude?
number-theory
asked Jan 4 at 13:14
LanceLance
8912
add a comment |
Given two positive integer numbers $m,n$, show that $frac{m}{n}$ can't have period $b-1$ in base $b$.
MY ATTEMPT: wlog $m<n$, then suppose $frac{m}{n}=0,a_1a_2...a_{b-1}$. Multiplying both sides by $b^{b-1}-1$ we obtain $frac{m(b^{b-1}-1)}{n}=a_1...a_{b-1}$ with $a_i in {0,1,2,...,b-1}$. How can I conclude?
number-theory
asked Jan 4 at 13:14
LanceLance
8912
Given two positive integer numbers $m,n$, show that $frac{m}{n}$ can't have period $b-1$ in base $b$.
MY ATTEMPT: wlog $m<n$, then suppose $frac{m}{n}=0,a_1a_2...a_{b-1}$. Multiplying both sides by $b^{b-1}-1$ we obtain $frac{m(b^{b-1}-1)}{n}=a_1...a_{b-1}$ with $a_i in {0,1,2,...,b-1}$. How can I conclude?
number-theory
number-theory
asked Jan 4 at 13:14
LanceLance
8912
asked Jan 4 at 13:14
LanceLance
8912
asked Jan 4 at 13:14
LanceLance
8912
asked Jan 4 at 13:14
LanceLance
8912
asked Jan 4 at 13:14
LanceLance
8912
8912
add a comment |
add a comment |
1 Answer
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I may miss something...
But $overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
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I may miss something...
But $overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
add a comment |
I may miss something...
But $overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
add a comment |
I may miss something...
But $overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
I may miss something...
But $overline{0,123456789123456789...}^{10}$ is a rational having a period equal to $9$ written in base $10$.
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
answered Jan 4 at 13:21
mathcounterexamples.netmathcounterexamples.net
25.3k21953
25.3k21953
add a comment |
add a comment |
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