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Let $a$ and $b$ same sign, $a,b\in \mathbb{Z}$ and $n\in\mathbb{N}$. Prove that $\exists c$ that the equation $ax+by=c$ has exactly n different positive solutions, it means $(x,y)$, where $x>0$ and $y>0$.

I have to use theorem:

Let $a,b,c\in\mathbb{Z}$. If at least one of the numbers $a$ and $b$ is not $0$ and $x_0, y_0$ is the equation $ax+by=c$ some solution, then all solutions $x,y$ of this equation is obtained by means of formulas $x=x_0+\frac{b}{gcd(a,b)}t$,
$y=y_0-\frac{a}{gcd(a,b)}t$, giving all integer values to the variable $t$.

First find a solution $x_0,y_0$ and then suitable $c$ value.

How to prove it?

Thank You.

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  • $\begingroup$ W.L.O.G we may assume that both $a,b\ge 0$. Then for $n=0$ we can take any $c<0$, so that $ax+by=c$ has exactly $0$ different positive solutions. So the claim is even true for $n=0$. $\endgroup$ – Dietrich Burde Feb 27 '18 at 12:35
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Let $d=\gcd(a,b)$ and $a'=a/d$, $b'=b/d$. Then

  • $x = x_0 + b't >0$ iff $t > -x_0/b'$

  • $y = y_0 - a't <0$ iff $t < y_0/a'$.

So, $t \in (-x_0/b', y_0/a')$. The length of this interval is $$ \frac{y_0}{a'}+\frac{x_0}{b'}=\frac{a'x_0+b'y_0}{a'b'}=\frac{c}{a'b'd}=\frac{c}{m} $$ where $m=lcm(a,b)$. Take $c=nm$ and the interval will have length $n$ and so $n$ integers inside it in general. If one or both extremes are integers, you'll have to compensate.

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$a x+b y =c$

1- We find gcd of a and b: gcd(a, b)

2- the condition for existence of solution is that : $gcd(a, b) | c$

3- we find the solutions of homogeneous equation $(ax+by=0)$; in case a and b are co-primes then $gcd(a, b)=1$ and we have:

$a x+b y = 0$:

$x= b$ and $y=-a$

So general solutions of homogeneous equation is:

$x = b . t$

$y = -a . t$

Where t can be any arbitrary value in Z. If we sum these solutions with a couple of certain solutions $x_0$,and $y_0$ then we find general solutions of equation $a x+b y =c$ ; we have:

$x = b.t + x_0$

$y = -a.t + y_0$

Hence this equation has infinitely many solutions in Z in general and in N in particular.The values of $x_0$ and $y_0$ depends on the value of c, therefore for n different c there can be n different sets of positive solutions.

Note that if a and b are co primes then their common divisor is $1$.

Geometric interpretation: Equation $ax +by=c$ in fact represent a line which is the locus of infinitely many points, among them many have coordinates in Z in general and in N in particular. With certain values of a and b , which define the slope of the lines c can have various values; each value gives a line parallel to others so that if we have n different c we will have n different equation or line with the same slope, hence we have exactly n solutions(or n lines) for the equation in each there exist many points having coordinates in N.So with n different c we have n different set of point having positive coordinates. c can be any arbitrary integer. in other words multiplying both sides of equation by a common divisor like $d$ we get a new equation( a new line).In this way all equations satisfy the condition that $gcd(ad, bd)=d|cd=c_1$ where $c_1$ is the new c.

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  • $\begingroup$ Where do we see that there are, for any given $n$, exactly $n$ different positive solutions? $\endgroup$ – Dietrich Burde Feb 27 '18 at 12:37
  • $\begingroup$ @Anette Uiga please see my edited answer. $\endgroup$ – sirous Feb 27 '18 at 15:42
  • $\begingroup$ @DietrichBurde, for any given n there exists infinitely many positive solutions(not n solutions).So for any given n there are exactly n different sets of positive solutions. $\endgroup$ – sirous Feb 28 '18 at 3:19
  • $\begingroup$ Infinitely many? Certainly not. Take $2x+4y=1$. $\endgroup$ – Dietrich Burde Feb 28 '18 at 9:21
  • $\begingroup$ @DietrichBurde, As I explained in geometric interpretation each n gives a new equation which has at least one positive solution. n can goes to infinity therefore there can be infinitely many positive solutions for this type of equation. $\endgroup$ – sirous Feb 28 '18 at 11:52

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