2.5 Prime Factorization and the Least Common Multiple

Prime Factorization and the Least Common Multiple

Prime factorization breaks any composite number down into the prime numbers that multiply together to make it. Think of it as finding the most basic building blocks of a number. This skill is essential for finding the least common multiple (LCM), which shows up constantly when working with fractions, scheduling problems, and packaging scenarios.

Prime Factorization

Basic Number Concepts

Before diving into prime factorization, make sure you're solid on these terms:

• Factor: A number that divides evenly into another number with no remainder. The factors of 12 are 1, 2, 3, 4, 6, and 12.
• Multiple: What you get when you multiply a number by a positive integer. The first few multiples of 5 are 5, 10, 15, 20, 25...
• Product: The result of multiplying two or more numbers together. The product of 3 and 7 is 21.
• Prime number: A number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, 13...).
• Composite number: A positive integer greater than 1 that has factors other than just 1 and itself (4, 6, 8, 9, 10, 12...). These are the numbers you can break down further.

Prime Factorization of Composite Numbers

Prime factorization means expressing a composite number as a product of prime factors. Every composite number has exactly one prime factorization (this is actually a famous math fact called the Fundamental Theorem of Arithmetic).

Steps to find the prime factorization:

1. Divide the number by the smallest prime that goes into it evenly.
2. Take the result and divide again by the smallest prime that works.
3. Keep dividing until you reach 1.
4. Write out all the prime divisors you used. That product is your prime factorization.

Example: Prime factorization of 24

• $24 \div 2 = 12$
• $12 \div 2 = 6$
• $6 \div 2 = 3$
• $3 \div 3 = 1$
• Prime factorization: $2 \times 2 \times 2 \times 3 = 2^3 \times 3$

Example: Prime factorization of 60

• $60 \div 2 = 30$
• $30 \div 2 = 15$
• $15 \div 3 = 5$
• $5 \div 5 = 1$
• Prime factorization: $2 \times 2 \times 3 \times 5 = 2^2 \times 3 \times 5$

Notice you always start with the smallest prime (2) and only move to a larger prime when the smaller one no longer divides evenly.

Least Common Multiple

Finding the LCM Using Prime Factorization

The least common multiple (LCM) of two or more numbers is the smallest positive integer that all of them divide into evenly.

Steps to find the LCM:

1. Find the prime factorization of each number.
2. For each prime factor that appears, take the highest power of that prime from either factorization.
3. Multiply those highest powers together. That's your LCM.

The key detail in step 2 is worth emphasizing: you don't just combine all the factors from both numbers. You compare and take the highest power of each prime.

Example: LCM of 12 and 18

• Prime factorization of 12: $2^2 \times 3$
• Prime factorization of 18: $2 \times 3^2$
• Highest power of 2: $2^2$ (from 12)
• Highest power of 3: $3^2$ (from 18)
• LCM: $2^2 \times 3^2 = 4 \times 9 = 36$

Example: LCM of 8 and 12

• Prime factorization of 8: $2^3$
• Prime factorization of 12: $2^2 \times 3$
• Highest power of 2: $2^3$ (from 8)
• Highest power of 3: $3$ (from 12)
• LCM: $2^3 \times 3 = 8 \times 3 = 24$

Applications of the LCM

LCM problems usually involve situations where two things happen in different-sized groups and you need to find when they line up evenly.

Bakery packaging: A bakery sells donuts in boxes of 12 and muffins in boxes of 18. What's the smallest number of each item you'd need so that you have the same total of donuts and muffins, all in full boxes? Find the LCM of 12 and 18, which is 36. You'd need 36 donuts (3 boxes) and 36 muffins (2 boxes).

Filling classrooms and buses: A school has classrooms that seat 24 students and buses that hold 36 students. What's the minimum number of students needed to completely fill some number of classrooms and completely fill some number of buses?

• Prime factorization of 24: $2^3 \times 3$
• Prime factorization of 36: $2^2 \times 3^2$
• Highest power of 2: $2^3$; highest power of 3: $3^2$
• LCM: $2^3 \times 3^2 = 8 \times 9 = 72$

You'd need 72 students (filling 3 classrooms and 2 buses).