Relationship among LCM, GCD, and coprimes

[andresc889]

Hi everybody,

I am having an extremely hard time understanding a specific relationship that originates from the least common multiple of two or more numbers.

According to "Number Theory and Its History" by Oystein Ore, it is not difficult to see that when one writes

lcm(a,b,c) = p*a = q*b = r*c

Then

gcd(p,q,r) = 1

Meaning that p, q, and r are relatively prime (coprimes).

I have verified this property with several examples. However, I feel somewhat stupid because I cannot see why this is true, and Ore is saying "it is not difficult to see."

According to Wikipedia,

"(...) if a and b are two rationals (or integers), there are integers m and n such that LCM(a, b) = m × a = n × b. This implies that m and n are coprime; otherwise they could be divided by their common divisor, giving a common multiple less that the least common multiple, which is absurd."

I am not able to see how this argument works. Can anybody help me understand how this property is proven?

Thank you in advance.

 

[Citan Uzuki]

Basically, it's saying that if m and n have a common divisor d>1, then (m/d)*a = (n/d)*b is also a common multiple of a and b. But (m/d)*a < m*a = LCM (a, b), so this is impossible. The same trick works with the three-variable version.

 

[andresc889]

Thank you Citan. It finally makes sense!