Find two rational numbers with denominators 11 and 13, respectively, and a sum of 7/143.
I got x/11 + y/13=7/143.
Two unknowns and one equation. Kind of stuck.
Don't forget and are integers. The equation they satisfy is .
You should first find one particular solution (you may start by finding such that (why is it possible ?)), and then look for the others: they satisfy , hence and you should then be able to conclude that and for some integer .
http://en.wikipedia.org/wiki/Euclidean_algorithm). Actually, do the Euclidian algorithm over 13 and 11, and observe...
Then, multiply by 7 :
Use the Euclidean Algorithm to get the gcd of 11 and 13 (yes, you know this is 1, but the calculations you did in the algorithm help you find out what values of a and b give you 11a + 13b = 1.
You now want two numbers x and y such that 11x + 13y = 7.
Well, you just got 11a + 13b = 1, so multiply everything by 7:
So 7a and 7b are the numbers you want for x and y. Job done.