Recall that when two or more numbers are multiplied, each number is a factor of the product. For example, 12 can be expressed as the product of different pairs of whole numbers. Factors can be shown geometrically.
The whole numbers 1, 12, 2, 6, 3, and 4 are the factors of
12.
Some whole numbers have exactly two factors, the number
itself and 1. Recall that these numbers are called prime numbers. Whole
numbers that have more than two factors, such as 12, are called composite
numbers.
Find the factors of each number. Then classify each number as
prime or composite.
1.
72
Alternative Solutions:
To
find the factors of 72, list all pairs of whole numbers whose product is 72.
The
factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Since 72 has more
than two factors, it is a composite number.
2.
37
Alternative Solutions:
There
is only one pair of whole numbers whose product is 37.
1 ×
37
The factors
of 37 are 1 and 37. Therefore, 37 is a prime number.
You can use a graphing calculator to investigate factor
patterns.
Since 4 ⋅ 3 = 12,
4 is a factor of 12. However, it is not a prime factor of 12 because 4 is not a
prime number. Recall that when a number is expressed as a product of prime
factors, the expression is called the prime factorization of the number.
You can use a factor tree to find the prime
factorization of a number. Two different factor trees are shown for the prime
factorization of 12.
All of the factors in the last row are prime numbers. The
factors are in a different order, but the result is the same. Except for the
order of the factors, there is only one prime factorization of a number. Thus,
the prime factorization of 12 is 2 ⋅ 2 ⋅ 3 or 22
⋅ 3.
You can use prime factorization to factor monomials. A
monomial is in factored form when it is expressed as the product of
prime numbers and variables and no variable has an exponent greater than 1.
Example
Factor each monomial.
3.
12a2b
Alternative Solutions:
4.
100mn3
Alternative Solutions:
5.
–25x2
Alternative Solutions:
To factor a negative integer, first express it as the product of a whole number and –1. Then find the prime factorization.
Two or more numbers may have some common prime factors.
Consider the prime factorization of 36 and 42.
The integers 36 and 42 have 2 and 3 as common prime factors.
The product of these prime factors, 2 ⋅ 3 or 6,
is called the greatest
common factor (GCF) of 36 and 42. The GCF is the greatest number
that is a factor of both original numbers.
The GCF of two or more monomials is the product of their
common factors when each monomial is expressed in factored form.
Example
Find the GCF of each set of numbers or monomials.
6.
24, 60, and 72
Alternative Solutions:
The
GCF of 24, 60, and 72 is 2 ⋅ 2 ⋅ 3 or
12.
7.
15 and 8
Alternative Solutions:
There
are no common prime factors. The only common factor is 1. So, the GCF of 15 and
8 is 1.
8.
15a2b and 18ab
Alternative Solutions:
The GCF of
15a2b and 18ab is 3 ⋅ a ⋅ b or 3ab.
Knowing the factors of a number can help you with geometry.
Geometry
Link
9. The area of a rectangle is 18 square inches. Find the length and width so
that the rectangle has the least perimeter. Assume that the length and width
are both whole numbers.
Alternative Solutions:
Sumber
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