Some Number Theory proofs about GCD and LCM

Question

how to prove or disprove the following :-

$$\gcd (kn,km) = k\gcd(n,m).$$

$$\operatorname{lcm}(n,m)\gcd(n,m)=mn.$$

$$\operatorname{lcm}(kn,km)=k\operatorname{lcm}(n,m).$$

It should probably be added that $m$, $n$, and $k$ are positive. If you already have and can use the Unique Factorization Theorem (aka the Fundamental Theorem of Arithmetic), then it comes down to comparing exponents of the primes. If this comes before Unique Factorization, more detailed work is involved.
What is your definition of $\operatorname{gcd}$? The first one is certainly true—can you try to prove it?
@Dylan I have it on good authority that his definition of gcd is taking the minimum exponent of every prime ;).

Gone · Answer 1 · 2012-01-16 16:27:50Z

THEOREM $\rm\;\; (a,b)\ =\ ab/[a,b] \;\;$ if $\;\rm\ [a,b] \;$ exists.

Proof $\rm\ \ d\ |\ a,b \;\iff\; a,b\ |\ ab/d \;\iff\; [a,b]\ |\ ab/d \;\iff\; d\ |\ ab/[a,b]\ \ $ QED

1 Answer