Show that only integral value of $frac{a}{b} + frac{b}{a}$ is $2$ when $a,b in Bbb{Z}$












4














I am given the equation
$$
m=frac{a}{b} + frac{b}{a}
$$

where $a$ and $b$ are positive integers. I have to show that the only integral value $m$ can take is $2$.



I have no idea how to solve it. I tried rewriting the equation as
$$
a^2 + b^2 - mab = 0
$$

But that tells me nothing.



Im stuck, please help!










share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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  • Are $a$ and $b$ integers?
    – 5xum
    yesterday










  • Yes, a and b are positive integers
    – Funny guy
    yesterday










  • Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
    – gandalf61
    yesterday
















4














I am given the equation
$$
m=frac{a}{b} + frac{b}{a}
$$

where $a$ and $b$ are positive integers. I have to show that the only integral value $m$ can take is $2$.



I have no idea how to solve it. I tried rewriting the equation as
$$
a^2 + b^2 - mab = 0
$$

But that tells me nothing.



Im stuck, please help!










share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.




















  • Are $a$ and $b$ integers?
    – 5xum
    yesterday










  • Yes, a and b are positive integers
    – Funny guy
    yesterday










  • Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
    – gandalf61
    yesterday














4












4








4


2





I am given the equation
$$
m=frac{a}{b} + frac{b}{a}
$$

where $a$ and $b$ are positive integers. I have to show that the only integral value $m$ can take is $2$.



I have no idea how to solve it. I tried rewriting the equation as
$$
a^2 + b^2 - mab = 0
$$

But that tells me nothing.



Im stuck, please help!










share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I am given the equation
$$
m=frac{a}{b} + frac{b}{a}
$$

where $a$ and $b$ are positive integers. I have to show that the only integral value $m$ can take is $2$.



I have no idea how to solve it. I tried rewriting the equation as
$$
a^2 + b^2 - mab = 0
$$

But that tells me nothing.



Im stuck, please help!







elementary-number-theory quadratics






share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited yesterday









Daniel R

2,45832035




2,45832035






New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked yesterday









Funny guyFunny guy

236




236




New contributor




Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Funny guy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • Are $a$ and $b$ integers?
    – 5xum
    yesterday










  • Yes, a and b are positive integers
    – Funny guy
    yesterday










  • Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
    – gandalf61
    yesterday


















  • Are $a$ and $b$ integers?
    – 5xum
    yesterday










  • Yes, a and b are positive integers
    – Funny guy
    yesterday










  • Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
    – gandalf61
    yesterday
















Are $a$ and $b$ integers?
– 5xum
yesterday




Are $a$ and $b$ integers?
– 5xum
yesterday












Yes, a and b are positive integers
– Funny guy
yesterday




Yes, a and b are positive integers
– Funny guy
yesterday












Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
– gandalf61
yesterday




Then your question title should say $a,b in mathbb{Z}^+$ instead of $a,b in mathbb{Z}$
– gandalf61
yesterday










2 Answers
2






active

oldest

votes


















7














Suppose we have factored out all common factors of $a$ and $b$ so that $a,b$ are relatively prime. Then



$$a^2+b^2-mab = 0$$



implies that $bmid a$ and $amid b$. Do you see how? Thus both are $1$.






share|cite|improve this answer





















  • What do you mean by a|b and b|a?
    – Funny guy
    yesterday






  • 1




    $amid b$ meaning that $a$ divides $b$
    – Olof Rubin
    yesterday










  • I don't see how you can conclude a divides b and vice versa
    – Funny guy
    yesterday








  • 2




    For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
    – Olof Rubin
    yesterday








  • 1




    Good answer +1.
    – Prakhar Nagpal
    yesterday



















3














With



$$a^2 - mab + b^2 = 0 tag{1}label{eq1}$$



consider $a$ to be a variable, with $b$ and $m$ being constant, so you have a quadratic equation in $a$. Using the quadratic formula to solve for $a$ gives that



$$a = cfrac{mb pm sqrt{m^2 b^2 - 4b^2}}{2} tag{2}label{eq2}$$



This simplifies to



$$a = bleft(cfrac{m pm sqrt{m^2 - 4}}{2}right) tag{3}label{eq3}$$



As $a$ and $b$ are positive integers, for $m$ to also be an integer means that $m^2 - 4$ must be a perfect square. For positive integers $n$, note that $left(n + 1right)^2 - n^2 = 2n + 1$ is $gt 4$ for $n ge 2$, meaning the difference of just $2$ consecutive squares is then too large, so $m le 2$, with $m = 2$ working. This gives that $a = b$.






share|cite|improve this answer























  • Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
    – Funny guy
    yesterday












  • @Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
    – John Omielan
    yesterday










  • @Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
    – John Omielan
    yesterday













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2 Answers
2






active

oldest

votes








2 Answers
2






active

oldest

votes









active

oldest

votes






active

oldest

votes









7














Suppose we have factored out all common factors of $a$ and $b$ so that $a,b$ are relatively prime. Then



$$a^2+b^2-mab = 0$$



implies that $bmid a$ and $amid b$. Do you see how? Thus both are $1$.






share|cite|improve this answer





















  • What do you mean by a|b and b|a?
    – Funny guy
    yesterday






  • 1




    $amid b$ meaning that $a$ divides $b$
    – Olof Rubin
    yesterday










  • I don't see how you can conclude a divides b and vice versa
    – Funny guy
    yesterday








  • 2




    For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
    – Olof Rubin
    yesterday








  • 1




    Good answer +1.
    – Prakhar Nagpal
    yesterday
















7














Suppose we have factored out all common factors of $a$ and $b$ so that $a,b$ are relatively prime. Then



$$a^2+b^2-mab = 0$$



implies that $bmid a$ and $amid b$. Do you see how? Thus both are $1$.






share|cite|improve this answer





















  • What do you mean by a|b and b|a?
    – Funny guy
    yesterday






  • 1




    $amid b$ meaning that $a$ divides $b$
    – Olof Rubin
    yesterday










  • I don't see how you can conclude a divides b and vice versa
    – Funny guy
    yesterday








  • 2




    For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
    – Olof Rubin
    yesterday








  • 1




    Good answer +1.
    – Prakhar Nagpal
    yesterday














7












7








7






Suppose we have factored out all common factors of $a$ and $b$ so that $a,b$ are relatively prime. Then



$$a^2+b^2-mab = 0$$



implies that $bmid a$ and $amid b$. Do you see how? Thus both are $1$.






share|cite|improve this answer












Suppose we have factored out all common factors of $a$ and $b$ so that $a,b$ are relatively prime. Then



$$a^2+b^2-mab = 0$$



implies that $bmid a$ and $amid b$. Do you see how? Thus both are $1$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered yesterday









Olof RubinOlof Rubin

1,212316




1,212316












  • What do you mean by a|b and b|a?
    – Funny guy
    yesterday






  • 1




    $amid b$ meaning that $a$ divides $b$
    – Olof Rubin
    yesterday










  • I don't see how you can conclude a divides b and vice versa
    – Funny guy
    yesterday








  • 2




    For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
    – Olof Rubin
    yesterday








  • 1




    Good answer +1.
    – Prakhar Nagpal
    yesterday


















  • What do you mean by a|b and b|a?
    – Funny guy
    yesterday






  • 1




    $amid b$ meaning that $a$ divides $b$
    – Olof Rubin
    yesterday










  • I don't see how you can conclude a divides b and vice versa
    – Funny guy
    yesterday








  • 2




    For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
    – Olof Rubin
    yesterday








  • 1




    Good answer +1.
    – Prakhar Nagpal
    yesterday
















What do you mean by a|b and b|a?
– Funny guy
yesterday




What do you mean by a|b and b|a?
– Funny guy
yesterday




1




1




$amid b$ meaning that $a$ divides $b$
– Olof Rubin
yesterday




$amid b$ meaning that $a$ divides $b$
– Olof Rubin
yesterday












I don't see how you can conclude a divides b and vice versa
– Funny guy
yesterday






I don't see how you can conclude a divides b and vice versa
– Funny guy
yesterday






2




2




For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
– Olof Rubin
yesterday






For instance we can write $a^2+b^2-mab = 0Leftrightarrow a^2 = b(ma-b)$ and since $b$ divides the right-hand side it must also divide the left-hand side
– Olof Rubin
yesterday






1




1




Good answer +1.
– Prakhar Nagpal
yesterday




Good answer +1.
– Prakhar Nagpal
yesterday











3














With



$$a^2 - mab + b^2 = 0 tag{1}label{eq1}$$



consider $a$ to be a variable, with $b$ and $m$ being constant, so you have a quadratic equation in $a$. Using the quadratic formula to solve for $a$ gives that



$$a = cfrac{mb pm sqrt{m^2 b^2 - 4b^2}}{2} tag{2}label{eq2}$$



This simplifies to



$$a = bleft(cfrac{m pm sqrt{m^2 - 4}}{2}right) tag{3}label{eq3}$$



As $a$ and $b$ are positive integers, for $m$ to also be an integer means that $m^2 - 4$ must be a perfect square. For positive integers $n$, note that $left(n + 1right)^2 - n^2 = 2n + 1$ is $gt 4$ for $n ge 2$, meaning the difference of just $2$ consecutive squares is then too large, so $m le 2$, with $m = 2$ working. This gives that $a = b$.






share|cite|improve this answer























  • Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
    – Funny guy
    yesterday












  • @Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
    – John Omielan
    yesterday










  • @Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
    – John Omielan
    yesterday


















3














With



$$a^2 - mab + b^2 = 0 tag{1}label{eq1}$$



consider $a$ to be a variable, with $b$ and $m$ being constant, so you have a quadratic equation in $a$. Using the quadratic formula to solve for $a$ gives that



$$a = cfrac{mb pm sqrt{m^2 b^2 - 4b^2}}{2} tag{2}label{eq2}$$



This simplifies to



$$a = bleft(cfrac{m pm sqrt{m^2 - 4}}{2}right) tag{3}label{eq3}$$



As $a$ and $b$ are positive integers, for $m$ to also be an integer means that $m^2 - 4$ must be a perfect square. For positive integers $n$, note that $left(n + 1right)^2 - n^2 = 2n + 1$ is $gt 4$ for $n ge 2$, meaning the difference of just $2$ consecutive squares is then too large, so $m le 2$, with $m = 2$ working. This gives that $a = b$.






share|cite|improve this answer























  • Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
    – Funny guy
    yesterday












  • @Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
    – John Omielan
    yesterday










  • @Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
    – John Omielan
    yesterday
















3












3








3






With



$$a^2 - mab + b^2 = 0 tag{1}label{eq1}$$



consider $a$ to be a variable, with $b$ and $m$ being constant, so you have a quadratic equation in $a$. Using the quadratic formula to solve for $a$ gives that



$$a = cfrac{mb pm sqrt{m^2 b^2 - 4b^2}}{2} tag{2}label{eq2}$$



This simplifies to



$$a = bleft(cfrac{m pm sqrt{m^2 - 4}}{2}right) tag{3}label{eq3}$$



As $a$ and $b$ are positive integers, for $m$ to also be an integer means that $m^2 - 4$ must be a perfect square. For positive integers $n$, note that $left(n + 1right)^2 - n^2 = 2n + 1$ is $gt 4$ for $n ge 2$, meaning the difference of just $2$ consecutive squares is then too large, so $m le 2$, with $m = 2$ working. This gives that $a = b$.






share|cite|improve this answer














With



$$a^2 - mab + b^2 = 0 tag{1}label{eq1}$$



consider $a$ to be a variable, with $b$ and $m$ being constant, so you have a quadratic equation in $a$. Using the quadratic formula to solve for $a$ gives that



$$a = cfrac{mb pm sqrt{m^2 b^2 - 4b^2}}{2} tag{2}label{eq2}$$



This simplifies to



$$a = bleft(cfrac{m pm sqrt{m^2 - 4}}{2}right) tag{3}label{eq3}$$



As $a$ and $b$ are positive integers, for $m$ to also be an integer means that $m^2 - 4$ must be a perfect square. For positive integers $n$, note that $left(n + 1right)^2 - n^2 = 2n + 1$ is $gt 4$ for $n ge 2$, meaning the difference of just $2$ consecutive squares is then too large, so $m le 2$, with $m = 2$ working. This gives that $a = b$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited yesterday

























answered yesterday









John OmielanJohn Omielan

1,19918




1,19918












  • Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
    – Funny guy
    yesterday












  • @Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
    – John Omielan
    yesterday










  • @Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
    – John Omielan
    yesterday




















  • Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
    – Funny guy
    yesterday












  • @Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
    – John Omielan
    yesterday










  • @Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
    – John Omielan
    yesterday


















Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
– Funny guy
yesterday






Why must m^2-4 be a perfect square? Secondly what's the point of doing (n+1)^2-n^2?
– Funny guy
yesterday














@Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
– John Omielan
yesterday




@Funnyguy Since $a$, $b$ and $m$ are stated to be integers, the equation (3) can only hold if $sqrt{m^2 - 4}$ is also an integer, i.e., a integer, which means that $m^2 - 4$ must be the square of an integer, which is called a "perfect square". My point for checking $left(n + 1right)^2 - n^2$ is to show that even for as small as $n = 3$, the difference from it squared and the integer just below it is more than $4$, showing that it can't be the value as the difference of squares increase, going from $5$ to $7$, then $9$, etc. Only $m = 2$ works, with the result following from this.
– John Omielan
yesterday












@Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
– John Omielan
yesterday






@Funnyguy For $m^2 - 4$ to be a perfect square means that $m^2 - 4 = k^2$, for some integer $k$. Thus, $m^2 = k^2 + 4$, and $m ge k + 1$. However, as the difference of even just $1$ more squared less the value below it is $5$ or more for $n = 3$, this can only be possible if $k = 0$. I'm sorry if this is somewhat confusing (it's still early morning for me), with it perhaps being easier for you to just try a few numbers to see for yourself. It would have been simpler, and perhaps better, to just stated that only $m = 2$ works.
– John Omielan
yesterday












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