Could someone help me prove the following:
I'm denoting the least common multiple as [a,b].
(a) whenever a divides k and b divides k, then [a,b] divides k.
(b) [a,b]=(ab)/(gcd(a,b)) if a,b are greater than zero.
Thanks for the help.
Jimmy
Could someone help me prove the following:
I'm denoting the least common multiple as [a,b].
(a) whenever a divides k and b divides k, then [a,b] divides k.
(b) [a,b]=(ab)/(gcd(a,b)) if a,b are greater than zero.
Thanks for the help.
Jimmy
You know that there exists two integers a and x such that: ax=k
You know that there exists two intergers b and y such that: by=k
Let us say that the least common multiple of a and b is n
Thus there exists two integers n and c such that: nc=a
Thus there exists two integers n and d such that: nd=b
Go back to: ax=k
thus: x=k/a
then: x=k/(nc)
multiply both sides by c: xc=k/n
Now, two integers multiplied together equal an integer, so k/n is an integer, thus k is divisible by n.