Download 1.4 Prime Factorization Example 1: Find all whole number factors of

Math 40
Prealgebra
Section 1.4 – Prime Factorization
1.4 Prime Factorization
Reminder:
In the statement 3 4  12 , the number 12 is called the product, while 3 and 4 are
called the factors.
Example 1: Find all whole number factors of 18.
Solution: We need to find all whole number pairs whose product equals 18. All pairs of numbers whose
product is 18 are
1 18  18 and 2 9  18 and 3 6  18
Hence, the factors of 18 (in order) are 1, 2, 3, 6, 9, and 18.
You Try It 1: Find all whole number factors of 21.
Divisible
Let a and b be whole numbers. Then a is divisible by b if and only if the remainder is zero
when a is divided by b. In this case, we say that “b is a divisor of a.”
Factors and Divisors
If c  a b , then a and b are called factors of c. Both a and b are also called divisors of c.
Note: Saying, “All whole number factors” is the same as saying, “All whole number
divisors.”
Example 2: Find all whole number divisors of 18.
Solution: Again, this is the same as asking, “find all whole number factors of 18.” Therefore, this question is
the same as Example 1. Hence, the divisors of 18 are 1, 2, 3, 6, 9, and 18.
You Try It 2: Find all whole number divisors of 24.
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2015 Worrel
Math 40
Prealgebra
Section 1.4 – Prime Factorization
Divisibility Tests
Divisible by 2: If a whole number ends in 0, 2, 4, 6, or 8, then the number is called an even
number and is divisible by 2.
Divisible by 3: If the sum of the digits of a whole number is divisible by 3, then the number
itself is divisible by 3.
EX: The number 141 is divisible by 3 because 1  4  1  6 and 6 is divisible
by 3. 141  3  47 
EX: The number 233 is not divisible by 3 because 2  3  3  8 and 8 is not
divisible by 3.
Divisible by 4: If the last two digits of a number is divisible by 4 then the entire number is
divisible by 4.
Note: If a number is not divisible by 2, it will not be divisible by 4.
EX: We can tell that 11,524 is divisible by 4 by looking at it, since its last
two digits, “24”, is divisible by 4. 11, 524  4  2,881
EX: We can tell that 13,815 is not divisible by 4 by looking at it, since its
last two digits, “15”, is not divisible by 4.
Divisible by 5: If the number ends in 0 or 5, then the number is divisible by 5.
Divisible by 6: If a number is divisible by 2 and 3, it is divisible by 6.
EX: The number 738 is divisible by 2 since it ends in an 8. Also, since
7  3  8  18 and 18 is divisible by 3, the number 738 is divisible by 3.
Hence since 738 is divisible by 2 and 3, it is also divisible by 6.
Divisible by 8: If the last three digits of a number is divisible by 8 then the entire number is
divisible by 8.
EX: We can tell that 73,032 is divisible by 8 by looking at it, since its last
three digits, “032”, is divisible by 8.  73, 032  8  9,129 
Divisible by 9: If the sum of the digits of a whole number is divisible by 9, then the number
itself is divisible by 9.
Prime Numbers
A whole number (other than 1) is a prime number if its only factors (divisors) are 1 and itself.
Equivalently, a number is prime if and only if it has exactly two factors (divisors).
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Math 40
Prealgebra
Section 1.4 – Prime Factorization
Example 3: Which of the whole numbers 12, 13, 21, and 37 are prime numbers?
Solution: 12: The factors (divisors) of 12 are 1, 2, 3, 4, 6, and 12. Hence, 12 is not prime.
13: The factors (divisors) of 13 are 1 and 13. Since there are only two factors (divisors), 1 and itself,
13 is a prime number.
21: The factors (divisors) of 21 are 1, 3, 7, and 21. Hence, 21 is not prime.
37: The factors (divisors) of 37 are 1 and 37. Since there are only two factors (divisors), 1 and itself,
37 is a prime number.
So, out of these four numbers, 13 and 37 are prime.
You Try It 3: Which of the whole numbers 15, 23, 51, and 59 are prime numbers?
Example 4: List all prime numbers less than 20.
Solution: The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19.
You Try It 4: List all prime numbers less than 40.
Composite Numbers
If a whole number is not prime, or 1, it is called a composite number.
Note: The number, 1, is neither prime nor composite.
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2015 Worrel
Math 40
Prealgebra
Section 1.4 – Prime Factorization
Example 5: Is the whole number 1,179 prime or composite?
Solution: We will try the prime numbers to see if the number is divisible by any of them.
2: 1,179 is not divisible by 2 since it does not end in 0, 2, 4, 6, or 8.
3: 1  1  7  9  18 , so the number, 1,179, is divisible by 3 and 9 since 18 is divisible by 3 and 9.
Hence, the number is composite.
You Try It 5: Is the whole number 2,571 prime or composite?
Factor Trees
We use factor trees to deconstruct numbers into a product of its prime number factors.
EX: Use a factor tree to find the prime factors of 18.
18
Break up 18 into the product of two factors
2
9
Keep breaking up any numbers that are
composite until you only have prime numbers
2
3
3
Hence, we can rewrite 18 as 18  2 3 3
Example 6: Express 24 as a product of prime factors.
Solution: Use a factor tree to find the prime factors.
24
2
4
6
2
2
3
We can rewrite 24 as 24  2 2 2 3
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2015 Worrel
Math 40
Prealgebra
Section 1.4 – Prime Factorization
You Try It 6: Express 36 as a product of prime factors.
Unique Factorization Theorem
Every whole number can be uniquely factored as a product of primes.
Exponents
The expression a n is defined to mean
an  a a
a
n times
The number, a, is called the base of the exponential expression and the number, n, is called the
exponent. The exponent, n, tells us to repeat the base, a, as a factor n times.
exponent
an
base
Example 7: Evaluate 25 , 34 , and 52 .
Solution:
25  2 2 2 2 2
34  3 3 3 3
 32
 81
52  5 5
 25
Hence, 2  32, 3  81, and 5  25 .
5
4
2
You Try It 7: Evaluate 35 and 23 .
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2015 Worrel
Math 40
Prealgebra
Section 1.4 – Prime Factorization
Example 8: Express the solution to Example 6 in compact form using exponents.
Solution: From Example 6 we had, 24  2 2 2 3 . Since there are multiple 2’s we can use exponents to express
it in compact form. The base is 2 and the exponent is 3 (since there are three 2’s).
Hence, 24  23 3 .
You Try It 8: Find the prime factors of 54. Write the answer in compact form using exponents.
Example 9: Evaluate the expression 23 32 52 .
Solution: Expand each factor raised to an exponent, then perform the multiplications.
2 3 32 5 2  2 2 2 3 3 5 5
8
9
25
 8 9 25
72
 72 25
 1,800
You Try It 9: Evaluate 33 52 .
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Math 40
Prealgebra
Section 1.4 – Prime Factorization
Area of a Square
Let s represent the length of each side of a square.
s
s
s
s
Since a square is also a rectangle, we can find the area of the square by using the area formula for
a rectangle, A  L W . However, in the case of the square, the length and width are the same
measurement, s. Therefore, A  s s  s 2 .
Hence, the formula for the area of a square is
A  s2
Example 10: The edge of a square is 13 centimeters. Find the area of the square.
Solution: Substitute s  13 cm into the area formula for a square, A  s 2 .
A  s2
 13 cm 
2
 13 cm 13 cm 
 169 cm 2 or 169 square centimeters
You Try It 10: The edge of a square is 15 centimeters. Find the area of the square.
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