Unit 2
Math 116
Prime numbers, Rational Numbers, and the
Pythagorean Theorem
Math 116
Section 2.2
Prime Numbers
A prime number is a natural number that has exactly two divisors.
A natural number that has more than two divisors is called a
composite number.
Examples of Prime
Numbers
5 is a prime number since its only factors are 1 and 5.
7 is prime number since its only factors are 1 and 7
Prime Factorization
of Numbers
Examples
Write the prime factorization of the given number or write the factor tree of the number
1) 24
2) 36
3) 90
Divisibility Rules
|
Number |
Divisibility Rule |
|
2 |
Last digit is divisible by 2 |
|
3 |
Sum of digits is divisible by 3 |
|
4 |
The last two digits are divisible by 4 |
|
5 |
The last digit is 0 or 5 |
|
6 |
The number is divisible by 2 and 3 |
|
8 |
The number formed by the last 3 digits is divisible by 8 |
|
10 |
The number is divisible by 2 and 5 Last digit is zero |
Divisibility statements
Page 158
12)
a) Does 6/39
False, 6 does not divide 39
b) Does 5/30
True, 5 divides 30 six times
c) Does 16/576
True, 16 divides 36
d) Does 3/7,823
False, the sum of the digits is not divisible by 3
Find the prime
factorization of the given number
4) 699
5) 740
Square Root test for prime numbers
The largest prime that divides a
number N must be less than or equal to
.
Examples
1) Determine if 301 is prime.
So start with 17 and check all numbers that are less than or equal 17.
If you check all numbers less than or equal to 17, it turns out that 301 is prime.
2) Determine if 183 prime
So start with
, and check all numbers less than or equal to 16.
After checking all factors less than or equal to 16, it turns out none of these numbers divide 283. Therefore, 283 prime.
3) Determine if 231 is prime.
Check the number 11
which divide 232 twenty one times. Therefore 232 is not a prime number.
Rational Numbers
A number that can be expressed as a quotient of integers
Examples
All of these numbers
can be expressed as a fraction
Irrational Numbers
A number that can not be expressed as a quotient of integers
Examples
Radicals and the
Pythagorean Theorem
Section 2.5
Radicals
Square Roots
Simplifying radicals
to lowest form
Rule for
multiplication of Radicals
Examples
Simplify each radical
(Optional)
1)
2)
3)
4)
5)
Pythagorean Theorem
In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
Examples
1)
2)
3)
How to determine if a triangle is a right triangle given the lengths of the sides
Converse of the Pythagorean Theorem
Let a, b, and c represent the three sides of a triangle, then:
The following is also true for acute and obtuse triangles
Let a, b, and c represent the three sides of a triangle, then:
Determine the type of triangle given the length of its sides
1)
2)
3)
Applications of the
Pythagorean Theorem
1) If a carpenter wants to make sure that a corner of a room is square and measures out 12 ft and 15 ft along the walls. How long should he make the diagonal?
1) An empty lot is 120 ft by 50 ft. How many feet would you save walking diagonally across the lot instead of walking length and width?
Compared to walking 120 ft + 50 ft = 170 ft
You would save walking 170 ft – 130 ft = 40 feet
2) A diagonal brace is to be placed in the wall of a room. The height of the wall is 10 feet and the wall is 24 feet long. What is the length of the brace?
4)
A television antenna is to be erected and held by guy wires. If the guy wires are 40 ft from the base of the antenna and the antenna is 50 ft high, what is the length of each guy wire?
