Prove that there is an infinite number of pairs $x$ and $y$ that satisfy $x+y=203$, and $\gcd(x,y)=7$

Question

I have a question.

Prove that there is an infinite number of pairs $x$ and $y$ that satisfy $$x+y=203, \text{ and } \gcd(x,y)=7$$

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If $\gcd(x,y)=7$, then $x=7p$, and $y=7q$.

then $$203=x+y=7(p+q)$$ $7$ divide $203$.

Hence such integers exist, but, how can I prove are infinite?

I could try with $\gcd(x,203-x)=7$.

Answer 1

If you begin with and easy example like $\space (x,y)=(7,196),\space$ and increase or decrease one variable by any multiple of $\space 7, \space$ while doing he opposite to the other variable, then $\quad GCD(x,y,203)=7, \space x \ne 0\space y \ne 0.$

Answer 2

Why didn't you simplify $203=x+y=7(p+q)$?

Dividing by $7$ we get $29 = p + q$. Now we are looking for $p$ and $q$ coprime.

But $\gcd(p,q) = \gcd(p,p+q) = \gcd(p,29)$, so we only have to choose $p$ coprime with $29$ (and since $29$ is prime, this is equivalent to choosing $p$ not multiple of $29$). Since $q = 29 - p$, the solutions are

$$x = 7 p$$ $$y = 7(29-p)$$

where $p$ is any number not multiple of $29$.