Question

Two rational numbers with denominators 11 and 13, respectively such that their sum is 7/143 is $-\frac{35}{11}$ and $\frac{42}{13}$
if true then enter $1$ and if false then enter $0$

Answer 1

Let the numbers be $\frac{x}{11}$ and $\frac{y}{13}$
Now,
$\frac{x}{11}+\frac{y}{13}=\frac{7}{143}$
$=>\frac{13x+11y}{143}=\frac{7}{143}$
$=>13x+11y=7$
Now, both $x$ and $y$ should be integers as denominator should be $11$ and $13$
Now,
$13x+11y=7$
$=>y=$$\frac{7-13x}{11}$
By, hit and trial, we find $x=9$ will give $y=-10$ and there can be many other solutions.
Therefore, the rational numbers are $\frac{9}{11}$ and $\frac{-10}{13}$