Abstract Alg: find two solutions in positive intergers for x^2-y^2=77?

Question

the only hint in the book is 77=1*77=7*11

and of course (x-y)*(x+y)

Answer 1

x^2-y^2=77

try a chart

let y = 1,2,3 etc. and see what x is.

y^2 = 77-x^2

nothing seems to work is all you want is integers

Answer 2

Factorise to get (x+y)(x-y)=77

As 77 = 1*77

x+y=77 and x-y=1,   solve to get x=39 and y=38

Using 77=7*11

x+y=11 and x-y=7 which gives x=9, y=2

Answer 3

u aldready have the answer!!! (x-y)(x+y)=7*11

only possibility is x-y=7 and x+y=11

implies x=9 y=2 !!! but still it took me 5 minutes to realise that!