Solve x + y = Relatively Prime: Need Help

[1+1=1]

i need some help here. i need to show that x=2k+1 and y=9k+4. i need to show that x and y are relatively prime. i am thinking of setting one = to k, and then substitiuting that into the otehr formula. any other suggestions?

 

[matt grime]

y-4x=k

hence d:=(x,y)|k

but d|x=2k+1 => d|1

 

[Gokul43201]

What matt means is the following (in case it needs clarification) :

Assume there is some number, d that divides y and x.

Then d should also divide any linear combination of y and x, in particular y-4x.

But since y-4x = k, this means that d divides k. So d should also divide 2k.

But since x=2k+1, d also divides 2k+1

The last two statements say that d divides 2k and 2k+1, which means that d should divide 1.

So, the only number that divides both x and y is 1.

 

[1+1=1]

o my that makes so much sense now. i was just setting one = to a k and then substitiuting it in the other. thank you both!