Least Common Multiple From GCD

Idea

When you know the greatest common divisor (GCD) of two positive integers, you can use it to find their least common multiple (LCM). The key relationship is that for two numbers, the product of the numbers equals the product of their GCD and LCM.

Method

1. Write the numbers and their GCD. Let the numbers be (a) and (b), and let (g = \gcd(a,b)).
2. Use the product relationship. Compute [ \operatorname{lcm}(a,b)=\frac{a\cdot b}{g}. ]
3. Simplify carefully. Divide before multiplying when possible to keep arithmetic easier.
4. State the result in simplest form. For integers, the LCM should also be a positive integer.

Check

A good check is to confirm that the result is a multiple of both original numbers and that the GCD condition is consistent. If you know one number and the GCD, the LCM should not be smaller than either number.

Common caution

Be sure the given GCD matches the two numbers. If the numbers are already coprime, the GCD is 1, so the LCM is just the product.