The LCM of two coprime numbers is equal to the product of the numbers.
Why the statement is true (step by step):
Coprime numbers are numbers that have no common factor other than 1. In symbols: \(\gcd(a,b)=1\).
There is an important fact that connects LCM and GCD:
\(\text{LCM}(a,b)\times \gcd(a,b)=a\times b\).
Because the numbers are coprime, \(\gcd(a,b)=1\). Put this into the fact above:
\(\text{LCM}(a,b)\times 1=a\times b\).
So, \(\text{LCM}(a,b)=a\times b\).
Quick example:
Take \(a=4\) and \(b=9\). They are coprime since \(\gcd(4,9)=1\).
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
Multiples of 9: 9, 18, 27, 36, …
The LCM is \(36\). Also, \(a\times b=4\times 9=36\).
Therefore, for coprime numbers, the LCM equals the product.