Lowest Common Multiple
The Lowest Common Multiple (LCM) of two numbers is the smallest positive number that is a multiple of both. Using prime factorization, the LCM is found by taking each prime factor the maximum number of times it appears in either number and multiplying them together. For example, the LCM of 8 and 14 is 56 because 8 = 2^3 and 14 = 2 x 7, so LCM = 2^3 x 7 = 56. This Grade 8 math skill from Yoshiwara Core Math Chapter 1 is essential for finding the LCD when adding fractions and appears in problems about recurring events or synchronized cycles.
Key Concepts
Property The lowest common multiple or LCM of two whole numbers is the smallest number that is a multiple of both whole numbers. To find the LCM using prime factorization, use each factor the most times it appears in either one of the two given numbers and multiply them together.
Examples To find the LCM of 9 and 12, list their multiples. Multiples of 9 are 9, 18, 27, 36, ... Multiples of 12 are 12, 24, 36, ... The smallest number on both lists is 36. Find the LCM of 10 and 15 using prime factorization. The factorizations are $10 = 2 \times 5$ and $15 = 3 \times 5$. We need one 2, one 3, and one 5. The $\text{LCM} = 2 \times 3 \times 5 = 30$. Let's find the LCM of 8 and 14. The prime factorizations are $8 = 2 \times 2 \times 2$ and $14 = 2 \times 7$. We need three 2s (from 8) and one 7 (from 14). The $\text{LCM} = 2 \times 2 \times 2 \times 7 = 56$.
Explanation The LCM is the smallest positive number that is a multiple of two or more numbers. It's useful for finding when two things with different cycles will happen at the same time, like buses arriving or gears aligning.
Common Questions
What is the lowest common multiple (LCM)?
The LCM of two or more numbers is the smallest positive number that is a multiple of all of them. For example, the LCM of 4 and 6 is 12 because 12 is the smallest number that both 4 and 6 divide into evenly.
How do you find the LCM using prime factorization?
Write the prime factorization of each number. For each prime factor, take the highest power that appears in any of the factorizations. Multiply all these together. For example, LCM of 10 and 15: 10 = 2 x 5 and 15 = 3 x 5. LCM = 2 x 3 x 5 = 30.
How do you find the LCM by listing multiples?
List multiples of each number until you find the same number in both lists. For example, to find LCM of 9 and 12: multiples of 9 are 9, 18, 27, 36... and multiples of 12 are 12, 24, 36... The first common multiple is 36.
When do 8th graders learn about LCM?
Students study the LCM in Grade 8 math as part of Chapter 1 of Yoshiwara Core Math, which covers preliminary ideas including number theory and factors.
What is the difference between GCF and LCM?
The GCF (Greatest Common Factor) is the largest number that divides both numbers. The LCM (Lowest Common Multiple) is the smallest number both divide into. For 12 and 8: GCF = 4 and LCM = 24.
What is a real-world use of the LCM?
The LCM is used when finding when two repeating events will coincide. If one event happens every 4 days and another every 6 days, they next coincide in LCM(4, 6) = 12 days.