I haven't been able to figure out this question for a while though I'm sure it's pretty straightforward. (It's intuitive to me and difficult to write into a proof). Show that these statements are equivalent
I) $\gcd(a,b)=c$ and $a+b=d$
II) $c|d$
I haven't been able to figure out this question for a while though I'm sure it's pretty straightforward. (It's intuitive to me and difficult to write into a proof). Show that these statements are equivalent
I) $\gcd(a,b)=c$ and $a+b=d$
II) $c|d$
If all you meant was "show that $I \implies II$" that is straightforward:
Let $c=\gcd(a,b)$. Write $a=cA$,$b=cB$. Then $$d=a+b=cA+cB=c(A+B)$$ so $c$ divides $d$.
But the other implication is not even remotely true. After all $1$ always divides $d$ yet $1$ isn't always the gcd of two specified numbers. Similarly, $d$ always divides $d$ but $d$ is larger than either $a$ or $b$, hence is not a divisor of either.