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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
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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

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