Least Common Multiple
Find the least common multiple of polynomials and integers in Grade 9 Algebra. Use prime factorization and factor trees to build the LCM efficiently. (Saxon Algebra 1, Grade 9)
Key Concepts
Property The least number that is evenly divisible by each of the numbers in a set of numbers. To find it, write each number as a product of prime factors. Then, use every prime factor the greatest number of times it appears in any single number's factorization.
Explanation Think of it like building the ultimate playlist from your friends' favorite songs, which are the prime factors. Your LCM playlist must include the maximum number of times any one friend listed a particular song. This way, everyone's top hit is covered, and the playlist is as short as possible while still pleasing the whole group.
Examples Find the LCM of 24 and 36. We have $24 = 2^3 \cdot 3$ and $36 = 2^2 \cdot 3^2$. The LCM needs the highest powers: $2^3 \cdot 3^2 = 72$. Find the LCM of 11, 12, and 18. We have $11=11$, $12 = 2^2 \cdot 3$, and $18 = 2 \cdot 3^2$. The LCM is $2^2 \cdot 3^2 \cdot 11 = 396$.
Common Questions
How do you find the LCM of two numbers using prime factorization?
Factor each number into primes. Take each prime factor at the highest power it appears in any of the numbers, then multiply these together. The result is the smallest number divisible by all given numbers.
Why is finding the LCM important in algebra?
The LCM is used as the least common denominator when adding or subtracting fractions and rational expressions. It lets you rewrite fractions with matching denominators so you can combine numerators directly.
How do you find the LCM of polynomial expressions?
Factor each polynomial completely. Identify all distinct factors and take each factor to the highest power it appears in any expression. Multiply these together to form the LCM of the polynomial expressions.