Prime Factorization and the Least Common Multiple

Prime Factorization and the Least Common Multiple

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Previous class

A number is divisible by 2 if the last digit is 0,2,4,6,or 8

3 if the sum of the digits is divisible by 3

5 if the last digit is 5 or 0

6 if divisible by both 2 and 3

10 if the last digit is 0

Questions

Is 127 prime or composite?

Is 810 divisible by 2,3,5,6,or 10?

Learning Objectives

• Find the prime factorization of a composite number

• Find the least common multiple (LCM) of two numbers

Find the Prime Factorization of a Composite Number

In the previous section, we found the factors of a number.

Prime numbers have only two factors, the number 1 and the prime number itself.

Composite numbers have more than two factors, and every composite number can be written as a unique product of primes.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

List of prime numbers less than 50

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Prime Factorization Using the Factor Tree Method

How To Find the prime factorization of a composite number using the tree method

Step 1. Find any factor pair of the given number, and use these numbers to create two branches.

Step 2. If a factor is prime, that branch is complete. Circle the prime.

Step 3. If a factor is not prime, write it as the product of a factor pair and continue the process.

Step 4. Write the composite number as the product of all the circled primes.

Find the prime factorization of 48

using the factor tree method.

Write the product of the circled numbers.

2⋅2⋅2⋅2⋅3

Write in exponential form. 2⁴⋅3

Prime Factorization Using the Factor Tree Method

we can write prime factorization inexponential form.

2⋅2⋅3⋅3

2²⋅3²

Find the prime factorization using the factor tree method: 80

Find the prime factorization using the factor tree method: 60

Find the prime factorization of 84 using the factor tree method.

Then we write 84 as the product of all circled primes.

2⋅2⋅3⋅7

2² ⋅3⋅7

Prime Factorization Using the Ladder Method

Prime Factorization Using the Ladder Method

The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.

To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for 36, we divide 36

by 2,the smallest prime factor of 36.

The quotient, 5, is prime, so the ladder is complete. Write the prime factorization of 120.

2⋅2⋅2⋅3⋅5

2³⋅3⋅5

Find the Least Common Multiple (LCM) of Two Numbers

Listing Multiples Method

A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of 10

and 25.

We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.

Least Common Multiple (LCM) of Two Numbers

10:10,20,30,40,50,60,70,80,90,100,110,…

25:25,50,75,100,125,…

How 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.

Find the LCM of 15 and 20 by listing multiples.

Solution

List the first several multiples of 15 and of 20. Identify the first common multiple.

15:15,30,45,60,75,90,105,120

20:20,40,60,80,100,120,140,160

The smallest number to appear on both lists is 60, so 60 is the least common multiple of 15 and 20.

Notice that 120 is on both lists, too. It is a common multiple, but it is not the least common multiple.

Multiple Choice

2 x 2 x 2 is the prime factorization of

• 6
• 12
• 22
• 8

Multiple Choice

Which list is only composite numbers?

• 1, 3, 4
• 2, 4, 6
• 4, 6, 8
• 5, 6, 8

Multiple Choice

Find the prime number.

• 10
• 9
• 7
• 4

Multiple Choice

Composite numbers:

• Are always bigger than prime numbers
• Have more than 2 factors
• Have 2 pairs of factors
• Don't have factors

Multiple Choice

Which of these is NOT a factor of 36?

• 2
• 8
• 12
• 6

Multiple Choice

All of these are prime except

• 3
• 9
• 2
• 13

Multiple Choice

Cups are sold 5 to a package and plates are sold 10 to a package. If you want to have the same number of each item for a party, what is the least number of packages of each you need to buy?

• 30
• 20
• 50
• 10

Multiple Choice

Choose the correct list of the first five multiples of 10.

• 1, 10, 20, 30, 40
• 10, 20, 30, 40, 50
• 5, 10, 15, 20, 25
• 10, 30, 40, 50, 60

Multiple Choice

What is the least common multiple (LCM) of 15 and 2?

• 30
• 2
• 45
• 152

Multiple Choice

What is the least common multiple (LCM) of 3 and 20?

• 60

• 40

• 20

• 3

Multiple Choice

What is the least common multiple (LCM) of 3 and 20?

• 60
• 40
• 20
• 3

Multiple Choice

If Juanita runs every third day and swims every fifth day, what is the first day that she will both run and swim again? 

• 5
• 15
• 45
• 30