Thread: gcd(a,b)=pi^min{si,ti} and lcm(a,b)=pi^max{si,ti} proof?

a>1 , b>1 are two integers(whole numbers)

s>=0 , t>=0

a=(P1^s1)*...*(Pr^sr)

b=(P1^t1)*...*(Pr^tr)

for each i=1,2,3,....,r is mi=min{si,ti} and ni=max{si,ti}.

show that : gcd(a,b)=(P1^m1)*...*(Pr^mr)


lcm(a,b)=(P1^n1)*...*(Pr^nr)




I've tried to ask the question as the way it is

Hope I'm clear enogh.

I would be glad if you explain the answers step by step.

Thanks in advance.

Use definition!
For example by definition
1) if g is a factor of a and b and i
2) if d is another factor of a and b then g divides d.

since then divides both and . Hence divides both a and b.

Now supposed d divides and let q be any of prime factor of d. Since d divides a then q must be one of . Then the prime factors of d are also . So we can write where . By doing the same thing with respect to b, we also have . Hence . Then it follows that g divides d.

The same (similar) argument also works for lcm.