+647 If a and b must be positive integers, what is the largest integer n such that 13a + 18b = n has no solutions?
+118673 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.
