PRE-ALGEBRA

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Lesson 4-1 Warm-Up

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Rules: The following divisibility rules are true for all numbers.

Example: Are 282, 468, 215, and 1,017 divisible by 3 or 9?

How can you tell if a number is divisible by 2, 3, 4, 5, 9, and 10?

Divisibility and Factors (4-1)

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Is the first number divisible by the second? Explain.

a. 1,028 by 2

Yes; 1,028 ends in 8.

b. 572 by 5

No; 572 doesn’t end in 0 or 5.

c. 275 by 10

No; 275 doesn’t end in 0.

Divisibility and FactorsLESSON 4-1

Additional Examples

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Is the first number divisible by the second?

a. 1,028 by 3

No; 1 + 0 + 2 + 8 = 11. 11 is not divisible by 3.

b. 522 by 9

Yes; 5 + 2 + 2 = 9. 9 is divisible by 9.

Divisibility and FactorsLESSON 4-1

Additional Examples

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Factors: the numbers you multiply together to get a product.Example: the product 24 has several factors.24 = 1 x 2424 = 2 x 1224 = 3 x 824 = 4 x 6

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24

To find the factors of a number:Start with 1 times the number.Try 2, 3, 4, etc.If you get doubles (such as 4 x 4), then you’re done. Repeats or doubles let you know you’re done.Example: What are the factors of 16?

3 isn’t a factor (doesn’t go into 16), so cross it out

Doubles or repeats mean your done!

The factors of 16 are 1, 2, 4, 8, and 16.

What is a “factor”?

How do you find the factors of a number?

1 x 16

2 x 8

3 x ?

4 x 4

Divisibility and Factors (4-1)

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Ms. Washington’s class is having a class photo taken.

Each row must have the same number of students. There are 35

students in the class. How can Ms. Washington arrange the

students in rows if there must be at least 5 students, but no

more than 10 students, in each row?

1 • 35, 5 • 7  Find pairs of factors of 35.

There can be 5 rows of 7 students, or 7 rows of 5 students.

Divisibility and FactorsLESSON 4-1

Additional Examples

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prime number: numbers that only have two factors: one, and the numberitselfExamples: 3, 5, 7, 11, 31

composite numbers: numbers that have more than two factors

Examples: 6, 15, 18, 30, 100

prime factorization: when a composite number is expressed as the product of prime numbers only

Example: 18 can be expressed as 3 x 3 x 2

Example: 40 can be expressed as 2 x 2 x 2 x 5

What are “prime numbers”?

What are “composite numbers:?

What are “composite numbers:?

Divisibility and Factors (4-1)

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61 has only two factors, 1 and 61. So 61 is prime.

a. 61

b. 65

Since 65 is divisible by 5, it has more than two factors. So 65 iscomposite.

Is each number prime or composite? Explain.

LESSON 4-1

Additional Examples

Divisibility and Factors

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2 x 50

To find the prime factorization of a number, make a factor tree as follows:.1.Write the product of a prime and composite number under the original number and draw lines connecting the factors with the original number.

2.Circle the prime number, and repeat step 1 with the composite factor.

3.Continue this process until the only numbers you have left are prime numbers.

4.Multiply all of the circled numbers together.

Example: What is the prime factorization of 100?

How do you find the prime factorization of a number?

100

2 x 25

5 x 5

2 is a prime numbers, so we are done with it.

5 is a prime numbers, so we are done with it.

So, the prime factorization of 100 is 2 x 2 x 5 x 5.

Divisibility and Factors (4-1)

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Since exponents show repeated multiplication (i.e. 34 means “3 x 3 x 3 x 3”), write any repeated prime numbers once and use an exponent to tell how many time that multiplication is repeated.

Example: In the previous example, we found the prime factorization of 100 as being 2 x 2 x 5 x 5.

2 x 2 can be expressed in exponent form as 22

5 x 5 can be expressed in exponent form as 52

So, 2 x 2 x 5 x 5 is more simply put as 22 x 52

How can we express prime factorization with exponents? “

Divisibility and Factors (4-1)

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prime 3 30

prime 3 10

prime 2 5

Find the prime factorization of 90.

The prime factorization of 90 is 2 • 3 • 3 • 5 or 2 • 32 • 5.

90

Stop when all factors are prime.

Use a factor tree. Because the sum of the digits of 90 is 9,90 is divisible by 3. Begin the factor tree with 3 • 30.

LESSON 4-1

Additional Examples

Divisibility and Factors

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State whether each number is divisible by 2, 3, 5, 9, or 10.

1. 18 2. 90 3. 81 4. 25

5. List the positive factors of 36.

2, 3, 9 2, 3, 5, 9, 10 3, 9 5

1, 2, 3, 4, 6, 9, 12, 18, 36

Lesson Quiz

Divisibility and FactorsLESSON 4-1