Lesson 4: Least Common Multiple (LCM) and Applications

Property

The smallest number that is a multiple of two numbers is called the least common multiple (LCM).
To find the least common multiple (LCM) of two numbers by listing multiples:
Step 1. List the first several multiples of each number.
Step 2. Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
Step 3. Look for the smallest number that is common to both lists.
Step 4. This number is the LCM.

Examples

• For 10 and 14: Multiples of 10 are $10, 20, 30, 40, ...$ and multiples of 14 are $14, 28, 42, ...$. The first common number is 70, so the $\operatorname{LCM}(10, 14) = 70$.
• For 9 and 15: Multiples of 9 are $9, 18, 27, 36, 45, ...$ and multiples of 15 are $15, 30, 45, ...$. The $\operatorname{LCM}(9, 15) = 45$.
• For 7 and 14: Multiples of 7 are $7, 14, 21, ...$ and multiples of 14 are $14, 28, ...$. The first one in common is 14, so the $\operatorname{LCM}(7, 14) = 14$.

Explanation

The LCM is the smallest positive number that is a multiple of two or more numbers. It's the first “meeting point” if you imagine listing out the multiples of each number in a sequence.

Property

To find the LCM using the prime factors method:
Step 1. Find the prime factorization of each number.
Step 2. Write each number as a product of primes, matching primes vertically when possible.
Step 3. Bring down the primes in each column.
Step 4. Multiply the factors to get the LCM.

Examples

• For 28 and 40: $28 = 2^2 \cdot 7$ and $40 = 2^3 \cdot 5$. The LCM needs the highest powers of all primes involved: $2^3 \cdot 5 \cdot 7 = 280$.
• For 20 and 30: $20 = 2^2 \cdot 5$ and $30 = 2 \cdot 3 \cdot 5$. Line up the factors and bring down the highest power of each prime: $2^2 \cdot 3 \cdot 5 = 60$.
• For 50 and 75: $50 = 2 \cdot 5^2$ and $75 = 3 \cdot 5^2$. The LCM is built from $2^1$, $3^1$, and $5^2$. So, $\operatorname{LCM}(50, 75) = 2 \cdot 3 \cdot 5^2 = 150$.

Explanation

This powerful method builds the LCM by taking every prime factor from all numbers at its highest power. This guarantees the result is divisible by all original numbers, making it the least common multiple.