Download 02-21 5.1 Divisibility Tests

§5.1 Divisibility Tests
02/19/17
Today We’ll Discuss
What does it mean for a number to be
divisible by another?
What are some common divisibility tests?
Definition of Divisibility
Definition: Let a and b be whole numbers
such that a ≠ 0. We say a divides b if and
only if there is a whole number n such that
b = an.
We write this as a∣b.
Definition of Divisibility
WAIT!! What does if and only if mean?
If a divides b then there exists a whole
number n such that b = an.
If there exists a whole number n such that
b = an then a divides b.
Definition of Divisibility
For a|b we can say:
• a is a factor of b
• b is a multiple of a
• b is divisible by a
• b is evenly divisible by a
(“Does not divide” is written as a∤b.)
Example
a) Use the definition of divisibility to state
why 3 divides 12.
b) Use the definition of divisibility to state
why 8 divides 1256.
Example
a) Give an example of a multiple of 12.
b) Give an example of a factor of 60.
c) True or False: 20∣4
d) Give an example of a number that is
divisible by 2 but not 4.
Divisibility Tests for 2, 5, 10
Test for 2: A whole number is divisible by 2
if and only if its ones digit is an even
number. (0,2,4,6,8)
Test for 5: A whole number is divisible by 5
if and only if its ones digit is 0 or 5.
Test for 10: A whole number is divisible by
10 if and only if its ones digit is 0.
Properties of Divisibility
a, b, m, and n are whole numbers (n ≠ 0)
Sum Property
If n∣a and n∣b, then n∣(a + b)
Difference Property
If n∣a and n∣b, then n∣(a − b)
Properties of Divisibility
a, b, m, and n are whole numbers (n ≠ 0)
Sum Property
If n∣a and n∣b, then n∣(a + b)
Difference Property
If n∣a and n∣b, then n∣(a − b)
Product Property
If n∣a, then n∣am
Example
Use the divisibility properties to complete
each statement.
a) 2∣4 and 2∣12, so …
b) 5∣15 and 5∣35, so…
c) 4∣32, so 4∣(5•__)
Developing Divisibility by 4
Do you know the divisibility test for 4?
Let’s take a look at how to discover it.
Divisibility Tests for 4, 8
Test for 4: A whole number is divisible by 4
if and only if the last two digits of the
whole number is divisible by 4.
Test for 8: A whole number is divisible by 8
if and only if the last three digits of the
whole number is divisible by 8.
Example
Determine whether or not the following are
divisible by 4 or 8:
a) 1450
b) 2728
c) 376
d) 1800
Developing Divisibility by 3, 9
Do you know the divisibility tests for 3 or
for 9?
Let’s take a look at how to discover them.
Divisibility Tests for 3, 6, 9
Test for 3: A whole number is divisible by 3
if and only if the sum of all its digits is
divisible by 3.
Test for 6: Passes tests for divisibility of
both 2 and 3.
Test for 9: A whole number is divisible by 9
if and only if the sum of all its digits is
divisible by 9.
Example
Determine whether or not the following are
divisible by 3, 6, or 9.
a) 1375
b) 2094
c) 3744
d) 1750
Discussion
If n∣(a + b), is it always true that n∣a and
n∣b?
Homework #10 §5.1
Pages 166-169
(Problems #7-21 You do not have to
explain but you should be able to on a
quiz or exam!)
#7,12,13,16,18,20,21,24,36,39