Lesson 4.1, For use with pages 175-180

ANSWER 4 ANSWER 15

2. 75 ÷ 51. 52 ÷ 13

Please do not use a calculator to solve these.

Lesson 4.1, For use with pages 175-180

2. 75 ÷ 51. 52 ÷ 13

Please do not use a calculator to solve these.

Divisibility Patterns and

Prime Factorization

Day 2

4.1

Vocabulary• Fundamental Theorem of Arithmetic: All

integers greater than one are prime or can be written as a unique product of prime numbers.

• Prime Factorization: the result of writing a number as a product of prime numbers. (use the factor tree method).

EXAMPLE 3 Writing Prime Factorization

Write the prime factorization of 450.

Three factor trees are shown. Notice that each factor tree produces the same prime factorization, differing only in the order of the factors.

So, 450 = 2 x 3 x 3 x 5 x 5.

Using exponents, the prime factorization of 450 is 2 x 32 x 52

GUIDED PRACTICE for Examples 2 and 3

6. 24

Tell whether the number is prime or composite. If it is composite, write its prime factorization using exponents.

So, 24 =

Number : 24

Factors : 1, 2, 3, 4, 6, 8, 12, 24

Composite; It has positive factors other then 1 and itself.

24

4 6

2 2 2 3.2 x 2 x 2 x 3 = 23 x 3

Find the prime factorization for the following numbers.

• 140

• 27

• 354

• 34

Find the prime factorization for the following numbers.

• 140– 22 x 5x 7

• 27– 33

• 354– 2 x 3 x 59

• 34– 2 x 17

What number do these prime factorizations represent?

• 2 x 33

• 22 x 3 x 11

• 2 x 3 x 5 x 11

What number do these prime factorizations represent?

• 2 x 33 – 54

• 22 x 3 x 11– 132

• 2 x 3 x 5 x 11– 330

Vocabulary• Monomial: a number, a variable, or a product of a

number and one or more variables.• Factor a Monomial: write it as a product of prime

numbers and variables with exponents of one.

EXAMPLE 4 Factoring a Monomial

Factor the monomial 12x2y.

Factor 12.

Write x2 as x x.

12x2y =

= 2 2 3 x x y

2 2 3 x2 y

GUIDED PRACTICE for Example 4

10. 3mn

Factor the monomial.

3mn = 3 m n

11. 18t2

Factor 18.18t2 = 2 3 3 t2

= 2 3 3 t t Write t2 as t t.

GUIDED PRACTICE for Example 4

13. 54w3z4

Factor the monomial.

54w3z4 = 2 3 3 3 w3 z4

= 2 3 3 3 w w w z z z z

Factor 54.

Write z4 as z z z z

Write w3 as w w w= 2 3 3 3 w w w z4

Homework

• Page #179 22-35, 52-53

EXAMPLE 1 Writing Factors

Members of the art club are learning to do calligraphy. Their first project is to make posters to display their new lettering style. A poster will display 36 characters in order: the 26 uppercase letters of the alphabet and the digits 0 through 9.

The art club members want the characters arranged in a rectangular display with the same number of characters in each row. You can use the factors of 36 to determine how many arrangements are possible.

Lettering

EXAMPLE 1 Writing Factors

In the situation above, how many ways can the art club arrange the 36 characters in a rectangular display with rows of equal length?

List the factors of 36 by writing 36 as a product of two numbers in all possible ways.

1 362 183 1264 9

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.