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1.1 Introduction to whole numbers  (Page 4/18)

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Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10.

by 2, 3, and 6

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Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10.

by 3 and 5

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Find prime factorizations and least common multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n , we can say that m is divisible by n . For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Since 8 · 9 = 72 , we say that 8 and 9 are factors    of 72. When we write 72 = 8 · 9 , we say we have factored 72.

An image shows the equation 8 times 9 equals 72. Written below the expression 8 times 9 is a curly bracket and the word “factors” while written below 72 is a horizontal bracket and the word “product”.

Other ways to factor 72 are 1 · 72 , 2 · 36 , 3 · 24 , 4 · 18 , and 6 · 12 . Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

Factors

If a · b = m , then a and b are factors    of m .

Some numbers, like 72, have many factors. Other numbers have only two factors.

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

Prime number and composite number

A prime number    is a counting number greater than 1, whose only factors are 1 and itself.

A composite number    is a counting number that is not prime. A composite number has factors other than 1 and itself.

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better understanding of prime numbers.

The counting numbers from 2 to 19 are listed in [link] , with their factors. Make sure to agree with the “prime” or “composite” label for each!

A table is shown with eleven rows and seven columns. The first row is a header row, and each cell labels the contents of the column below it. In the header row, the first three cells read from left to right “Number”, “Factors”, and “Prime or Composite?” The entire fourth column is blank. The last three cells read from left to right “Number”, “Factor”, and “Prime or Composite?” again. In each subsequent row, the first cell contains a number, the second contains its factors, and the third indicates whether the number is prime or composite. The three columns to the left of the blank middle column contain this information for the number 2 through 10, and the three columns to the right of the blank middle column contain this information for the number 11 through 19. On the left side of the blank column, in the first row below the header row, the cells read from left to right: “2”, “1,2”, and “Prime”. In the next row, the cells read from left to right: “3”, “1,3”, and “Prime”. In the next row, the cells read from left to right: “4”, “1,2,4”, and “Composite”. In the next row, the cells read from left to right: “5”, “1,5”, and “Prime”. In the next row, the cells read from left to right: “6”, “1,2,3,6” and “Composite”. In the next row, the cells read from left to right: “7”, “1,7”, and “Prime”. In the next row, the cells read from left to right: “8”, “1,2,4,8”, and “Composite”. In the next row, the cells read from left to right: “9”, “1,3,9”, and “Composite”. In the bottom row, the cells read from left to right: “10”, “1,2,5,10”, and “Composite”. On the right side of the blank column, in the first row below the header row, the cells read from left to right: “11”, “1,11”, and “Prime”. In the next row, the cells read from left to right: “12”, “1,2,3,4,6,12”, and “Composite”. In the next row, the cells read from left to right: “13”, “1,13”, and “Prime”. In the next row, the cells read from left to right “14”, “1,2,7,14”, and “Composite”. In the next row, the cells read from left to right: “15”, “1,3,5,15”, and “Composite”. In the next row, the cells read from left to right: “16”, “1,2,4,8,16”, and “Composite”. In the next row, the cells read from left to right, “17”, “1,17”, and “Prime”. In the next row, the cells read from left to right, “18”, “1,2,3,6,9,18”, and “Composite”. In the bottom row, the cells read from left to right: “19”, “1,19”, and “Prime”.

The prime number     s less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.

A composite number can be written as a unique product of primes. This is called the prime factorization    of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime factorization

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

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!

If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

How to find the prime factorization of a composite number

Factor 48.

Solution

This figure is a table that has three columns and four rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions and some math. The third column contains most of the math work corresponding with the written steps and instructions. In the top row, the first cell says: “Step 1. Find two factors whose product is the given number. Use these numbers to create two branches.” The second cell contains the algebraic equation 48 equals 2 times 24. In the third cell, there is a factor tree with 48 at the top. Two branches descend from 48 and terminate at 2 and 24 respectively. One row down, the instructions in the first cell say: “Step 2. If a factor is prime, that branch is complete. Circle the prime.” In the second cell, the instructions say: “2 is prime. Circle the prime.” In the third cell, the factor tree from step 1 is repeated, but the 2 at the bottom of the tree is now circled. One row down, the first cell says: “Step 3. If a factor is not prime, write it as the product of two factors and continue the process.” In the second cell, the instructions say: “24 is not prime. Break it into 2 more factors.” The third cell contains the original factor tree, with 48 at the top and two downward-pointing branches terminating at 2, which is underlined, and 24. Two more branches descend from 24 and terminate at 4 and 6 respectively. One line down, the instructions in the middle of the cell say “4 and 6 are not prime. Break them each into two factors.” In the cell on the right, the factor tree is repeated once more. Two branches descend from the 4 and terminate at 2 and 2. Both 2s are circled. Two more branches descend from 6 and terminate at a 2 and a 3, which are both circled. The instructions on the left say “2 and 3 are prime, so circle them.” In the bottom row, the first cell says: “Step 4. Write the composite number as the product of all the circled primes.” The second cell is left blank. The third cell contains the algebraic equation 48 equals 2 times 2 times 2 times 2 times 3.


We say 2 · 2 · 2 · 2 · 3 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer!

If we first factored 48 in a different way, for example as 6 · 8 , the result would still be the same. Finish the prime factorization and verify this for yourself.

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Find the prime factorization of 80.

2 · 2 · 2 · 2 · 5

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Find the prime factorization of 60.

2 · 2 · 3 · 5

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Find the prime factorization of a composite number.

  1. Find two factors whose product is the given number, and use these numbers to create two branches.
  2. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree.
  3. If a factor is not prime, write it as the product of two factors and continue the process.
  4. Write the composite number as the product of all the circled primes.
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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