The LCM in mathematics stands for `least common multiple`. The LCM indicates the lowest common multiple of 2 or more numbers. Common Applications of lcm can be found when operating on fractions via [[lcd]]. Example: > [!NOTE] > Problem: `lcm(4, 6)` > Multiples of 4: 4, 8, `12`, 16, 20, ... > Multiples of 6: 6, `12`, 18, 24, ... > Smallest Common Multiple: `12` Absolutely! Here's your **updated LCM note** with **two examples**—one with two numbers and one with **three numbers**—using prime factorization: ## Calculating LCM Calculating the LCM can be done by listing out all the multiples of each number & then finding the common one. This, however, is time-consuming for more than 2 inputs, especially as the smallest common multiple can be a large number. **Calculating Via Prime Factorization:** As explained in [[lcd]], you can easily figure out the LCM by calculating the prime factors of all the numbers, picking the **highest power** for each **unique prime factor**, and taking the **product**. The LCM is calculated in the exact same way! **Two Numbers:** > [!TIP] > Find `LCM(12, 18)` > > - $12 = 2^2 \cdot 3$ > > - $18 = 2 \cdot 3^2$ > > > Take the highest powers: > > - $2^2$, $3^2$ > > > **LCM = $2^2 \cdot 3^2 = 4 \cdot 9 = 36$** **Three Numbers:** > [!TIP] > Find `LCM(8, 9, 21)` > > - $8 = 2^3$ > > - $9 = 3^2$ > > - $21 = 3 \cdot 7$ > > > Collect all prime factors with their **highest power**: > > - $2^3$ (from 8) > > - $3^2$ (from 9) > > - $7$ (from 21) > > > **LCM = $2^3 \cdot 3^2 \cdot 7 = 8 \cdot 9 \cdot 7 = 504$** > [!NOTE] > The lcm & [[lcd]] also closely link to the [[hcf|hcf - highest common factor]]. > see also: [[hcf-lcm product identity]] ## Practice Tests - [[lcm-and-hcf-using-product-of-primes-pdf.pdf]]