To show $(n,n+1)=1$ for all $n>0$

Question

To prove $(n,n+1)=1$ for all $n>0$

i have tried using induction. So finally i have $(k,k+1)=1$ and i have to prove $(k+1,k+2)=1$

Let $(k+1,k+2)=d$. So $d|(k+1)$. Also $1|(k+1)$. Since $1$ and $d$ are both greatest numbers to divide $k+1$ So $1=d$

is this correct

Answer 1

That doesn't work, since $1$ is not in fact the greatest number to divide $k+1$. The simple solution is that $1=(n+1)-n$, which immediately implies $(n,n+1)=1$ since the gcd is the smallest positive integer you can obtain as a linear combination of the two numbers with (signed) integer coefficients.

Answer 2

You don't need induction.

Hint:

If $d\mid n$ and $d\mid n+1$ then also $d\mid (n+1)-n=1$.

Answer 3

No, induction doesn't work.

What you can do is this:

$$(n,n+1)=(n, \quad1\cdot (n+1)-1\cdot n)=(n,1)=1$$