Divisibility and Factors

4.1 p. 178

What do we want to accomplish?

• Review basic divisibility rules to help identify factors.

• Learn to write factors as “factor pairs.”• Understand why perfect squares have an odd

number of factors.

Basic Divisibility Rules

Divisibility Rules for 2, 5, and 10An integer is divisible by if it ends in 0, 2, 4, 6, or 8. if it ends in 0 or 5. if it ends in 0. numbers end in 0, 2, 4, 6, or 8 and are divisible

by . numbers end in 1, 3, 5, 7, or 9 and are not

divisible by 2.

2 5 10

Even 2

Odd

3

Divisibility Rules for 3 and 9An integer is divisible by if the sum of its digits is divisible by 3. if the sum of its digits is divisible by 9.9

One integer is divisible by another if ……

it divides without a remainder.

One integer is a factor of another integer if . . . it divides that integer with a remainder of 0.

Add two more rules to your notes:

A number is divisible by 6 if it is divisible by both 2 and 3A number is divisible by 4 if the last two digits are divisible by 4.

245 : 2 + 4 + 5 = 11 Not divisible by 3 or 9

255 : 2 + 5 + 5 = 12 Divisible by 3; not 9

Multiples are divisible by all of their factors!

Checking for Divisibility (Look on the back of the notes.)

Number 2 3 4 5 6 9 10 Sum of the digits

1,110

356

300

1028

572

275

2118

444

4000

1101

5220

√ √√ 3√ √

√ 14√

√ √ √ 3√ √√

√ 11√√ 14√

√ 14√ 12√ √

√ 12√ √√

√ √ √ 4√

3√

√ √ √ 9√ √ √√

Work vertically, one rule at a time. This focuses on the rule!

Fill in the rest of the digit sums before we

go on.

Listing all Factors of Numbers

Go back to page 1 of your notes.

All factors must be listed in pairs. Do not try to list them in order from smallest to largest. Your pair list will do that for you.

The technique we will use will help you factor polynomials next year.

96 100 84 1201, 96

2, ___

3, 32

4, ___

6, ___

8, 12

16

24

48

Reading down and up will give you your sequential list.

1,100

2, 50

4, 25

5, 20

10

1, 84

2, 423, ___

4, 21

28

6, 14 ? Use the 6 & 42

7, 12

1, 1202, 60

3, 404, 30

5, 24

6, 208, 15

10, 12

One more . . . . On the back of your notes…

You have 48 darling children in your art class. You want to arrange them in rows.

You want more than three rows, but the narrow room won’t allow for more than 6 rows. What configurations could you use?

48: 1, 48 2, 24 3, 16 4, 12 6, 8

You could have 4 rows of 12 students or 6 rows of 8 students.

I am well aware that some of you strongly RESIST doing something ina new format. TRY to write your factors in pairs. You will NEVER leaveone out if you use this method! You will also discover nice things aboutnumbers!

USE PAIRS to write factors.

34 50 36

1, 34

2, 17

1, 50

2, 255, 10

1, 362, 183, 124, 9

6

40 42 48

1, 402, 204, 105, 8

1,422,213,146, 7

1, 482, 243, 164, 126, 8

Why think in pairs???Next year, one of your skills will be to factor polynomials.

The skills of integer rules and factor pairs are fundamental to understanding this process.

48: 1, 48 2, 24 3, 16 4, 12 6, 8

Towards the end of the year, we will work witha process called “foil” that is part of this process.

This polynomial can be written as a productof two binomials.

What was the objective?

• Did you refresh your thinking about divisibility rules?

• Will you be able to write factors of a number in pairs?

• Could you explain why perfect squares have an odd number of factors?