indeterminate equation, JMO

Find the fractions, having 7 and 11 for their denominators, such that their sum is 1(34/77)




Answer:

x/7 + y/11 = 111/77

or, 11x + 7y = 111

or, 11(x + y) = 111 + 4y

 therefore, 111 + 4y must be a multiple of 11
Also, given x, y E integer,

So, 111 + 4y = 11n, where n E Z

or, y = (11n - 111) / 4

n               y                              x
11           fraction
12           fraction
13             8                        = (13 - 8) = 5
14           fraction

so, 5/7 + 8/11 = (55 + 56)/ 77 = 111/77

so, answer: x = 5 and y = 8