A factor is an integer that divides another integer evenly.
If a/b is
an integer, then b is a factor of a.
3, 4, and 6, for example, are factors of 12.
Sometimes it is necessary or helpful to factor an integer
completely. This means finding all the factors of that integer.
It’s possible that the test will directly require this skill or
will make use of it in a more complicated question. In either case,
it’s something you should know how to do.
To find all the factors of a number, write them down in
pairs, beginning with 1 and the number you’re factoring. We’ll factor
24 for this example. So 1 and 24 are both factors of 24. Next, try
every integer greater than 1 in increasing order. Here are the factor
pairs we find for 24:
- 1 and 24 (124
- 2 and 12 (212 = 24)
- 3 and 8 (38 = 24)
- 4 and 6 (46 = 24)
You know you’ve found all the factors of a number when
the next first factor exceeds its corresponding second factor. For
example, after you found that 4 was a factor of 24 and 5 was not,
you would see that 6, the next factor of 24, had already been included
in a pair of factors. Thus, all the factors have been found.
A prime number is a number whose only factors are 1 and
itself. All prime numbers are positive (because every negative number
has –1 as a factor in addition to 1 and itself). Furthermore, all
prime numbers besides 2 are odd. The first few primes, in increasing
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, . . .
To determine whether a number is prime, you shouldn’t
check whether the number is divisible by every number less than
itself. Such an effort would take an incredible amount of time,
and you have only an hour for the Math IIC. Instead, to decide whether
a number is prime, all you need to do is estimate the square root
of the number, then check all the prime numbers that fall below
your estimate. For example, to see if 91 is prime, you should estimate
the square root of the number:
Now you should test 91 for divisibility by the prime numbers smaller
than 10: 2, 3, 5 and 7.
- Is 91 divisible by 2? No, it does not end
with an even number.
- Is 91 divisible by 3? No, 9 + 1 = 10, and 10 is not divisible
- Is 91 divisible by 5? No, 91 does not end with 0 or 5.
- Is 91 divisible by 7? Yes! 917
Therefore, 91 is not prime.
Another form of factorization is called prime factorization.
The prime factorization of an integer is the listing of the prime
numbers whose product is that number.
To find the prime factorization of a number, divide it
and all of its factors until every remaining integer is prime. This
group of prime numbers is the prime factorization of the original
integer. Let’s find the prime factorization of 36 as an example.
It can be helpful to think of prime factorization in the
form of a tree:
As you may already have noticed, there is more than one
way to find the prime factorization of a number. We could have first
resolved 36 into 6
6, for example,
and then determined the prime factorization from there. So don’t
worry—you can’t screw up. No matter which path you take, you will
always get the same result. That is, as long as you do your arithmetic
correctly. Just for practice, find the prime factorizations for
45 and 41.
Since the only factors of 41 are 1 and 41, 41 is a prime
number. It is therefore its own prime factorization.
Greatest Common Factor
The greatest common factor (GCF) of two numbers is the
greatest factor that they have in common. Finding the GCF of two
numbers is especially useful in certain applications, such as manipulating
fractions. We’ll explain why later in this section.
In order to find the greatest common factor of two numbers,
we must first produce their prime factorizations. What
is the greatest common factor of 18 and 24, for example?
First, their prime factorizations:
The greatest common factor is the greatest integer that
can be written as a product of common prime factors. That is to
say, the GCF is the “overlap,” or intersection, of the two prime
factorizations. In this case, both prime factorizations contain
3 = 6. This is their GCF.
Here’s another example:
is the GCF of 96 and 144?
So, the product of the prime factors that they share is
= 48, which is their GCF.
For practice, find the GCF of the following pairs of integers:
12 and 15
Compare your answers to the solutions:
12 = 22 3.
15 = 3 5. The GCF is 3.
= 2 3 5.
45 = 32 5.
The GCF is 3 5 = 15.
= 1 13. 72 = 23 2.
There are no common prime factors. The GCF is 1.
= 2 7. 49 = 72.
The GCF is 7.
= 22 52.
80 = 24 5.
The GCF is 22 5
Relatively Prime Numbers
Two numbers are called relatively prime if they have no
common prime factors (i.e., if their GCF is 1). This doesn’t mean,
however, that each number is itself prime. 8 and 15 are relatively
prime because they have no common primes in their prime factorizations
(8 = 2
and 15 = 3
5), but neither
number is prime. It might be a good idea to know the definition
of relatively prime numbers, in case it pops up somewhere on the