Thread: coprime

how do you show that 2 consecutive integers are coprime?

i know we need to show that gcd is 1 where they have opp parity

Let your numbers be a and a+1 and their gcd be d.
$\displaystyle \Rightarrow d | a $ and $\displaystyle d | a+1 \Rightarrow d | (a+1-a) \Rightarrow d | 1 \Rightarrow d=1$

To show that two integers $\displaystyle m,n$ are coprime, it is sufficient to find integers $\displaystyle r,s$ such that $\displaystyle rm+sn=1$ (because any common divisor of $\displaystyle m,n$ must divide $\displaystyle rm+sn$ for all integers $\displaystyle r,s).$ In the case $\displaystyle m=a,$ $\displaystyle n=a+1,$ you have $\displaystyle n-m=1;$ hence $\displaystyle \gcd(m,n)=1.$