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**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