+0  
 
0
314
1
avatar+644 

If a and b must be positive integers, what is the largest integer n such that 13a + 18b = n has no solutions?

waffles  Aug 23, 2017
 #1
avatar+92775 
0

If a and b must be positive integers, what is the largest integer n such that 13a + 18b = n has no solutions?


I I don't know, I am getting really confused here . ://

 

This is what I am considering.

 

I used the euclidean algorithm, followed by the extended euclidean algorithm to come up with the generic solutions of

 

a= 7n-18k

b= -5n+13k

Where k is any integer.  So they are the answers to the diaphantine equation.   13a+18b=n

 

since a and be must both be positive

\(7n-18k>0\\ n>\frac{18k}{7}\qquad \qquad n>\frac{90k}{35} \\ -5n+13k>0\\ n<\frac{13k}{5}\qquad \qquad n<\frac{91k}{35} \\~\\ \frac{90k}{35} < n<\frac{91k}{35} \)

 

Mmm...   I have no idea where this is going. 

Melody  Aug 23, 2017
edited by Melody  Aug 23, 2017
edited by Melody  Aug 23, 2017
edited by Melody  Aug 23, 2017

13 Online Users

avatar
avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.