Transcript Divisibility Rules

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 10
An integer is divisible by
2 if it ends in 0, 2, 4, 6, or 8.
5 if it ends in 0 or 5.
10 if it ends in 0.
Even numbers end in 0, 2, 4, 6, or 8 and are divisible
by 2 .
Odd numbers end in 1, 3, 5, 7, or 9 and are not
divisible by 2.
Divisibility Rules for 3 and 9
An integer is divisible by
if the sum of its digits is divisible by 3.
3
255 : 2 + 5 + 5 = 12 Divisible by 3; not 9
9
if the sum of its digits is divisible by 9.
245 : 2 + 4 + 5 = 11 Not divisible by 3 or 9
Add two more rules to your notes:
A number is divisible by 6 if it is divisible by both 2 and 3
A number is divisible by 4 if the last two digits are divisible by 4.
One integer is divisible by another if ……
it divides without a remainder.
Multiples are divisible
by all of their factors!
One integer is a factor of another integer if . . .
it divides that integer with a remainder of 0.
Checking for Divisibility
Number
1,110
356
2
3
√
√
1028
√
√
5
√
√
300
572
4
6
√
9
Fill in the rest of the
digit sums before we
10go on. Sum of
the digits
√
√
√
√
√
√
√
√
√
√
444
√
√
4000
√
14
1101
√
√
√
√
√
3
11
√
2118
3
14
√
√
275
5220
(Look on the back of the notes.)
√
14
12
√
12
√
√
4
3
√
√
√
√
√
9
Work
vertically,
one rule at
a time. This
focuses on
the rule!
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
1, 96
1,100
1, 84
2, 50
2, 42
3, ___
28
2, ___
48
3, 32
4, ___
24
6, ___
16
4, 25
5, 20
10
84
4, 21
6, 14 ?
120
1, 120
2, 60
3, 40
4, 30
5, 24
Use the 6 & 42
7, 12
8, 12
Reading down and up will give you your sequential list.
6, 20
8, 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 in
a new format. TRY to write your factors in pairs. You will NEVER leave
one out if you use this method! You will also discover nice things about
numbers!
USE PAIRS to write factors.
34
50
1, 34
1, 50
2, 17
2, 25
5, 10
36
40
42
48
1, 36
2, 18
3, 12
4, 9
6
1, 40
2, 20
4, 10
5, 8
1,42
2,21
3,14
6, 7
1, 48
2, 24
3, 16
4, 12
6, 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 ofcan
thebe
year,
we will
with
This polynomial
written
as awork
product
a process
called “foil” that is part of this
of two binomials.
process.
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?