If $m$ is a positive integer, show that $3m+2$ and $5m+3$ are relatively prime

Question

I tried proving it by assuming the opposite. So (3m+2 , 5m+3)= k , k>1 3m+2=ka ; 5m+3=kb;

5m+3=3m+2+2m+1; 5m+3=ka + 2m+1; kb=ka +2m+1; 2m+1=kb-ka; 2m+1= 5m+3-3m+2; 2m+1=2m+1; Which means that they arent relatively prime , but if you test this with numbers you can clearly see that they are. What am i doing wrong ?

Answer 1

You just proved that $2m+1=2m+1$.

Try this (Euclidean algorithm) to show the gcd is $1$:

$$5m+3=1(3m+2)+(2m+1)$$

$$3m+2=1(2m+1)+(m+1)$$

$$2m+1=1(m+1)+m$$

$$m+1=1(m)+1$$

Answer 2

$$(5m+3;3m+2)=(2m+1;3m+2)=(2m+1;m+1)=(m;m+1)=1$$

Answer 3

If $d$ divides both $3m+2$ and $5m+3$, it must also divide $5(3m+2)-3(5m+3)=1$.