Factoring Numbers
written by Elizabeth Stapel
"Factors" are the numbers you multiply to get another number. For instance, the factors of 15
are 3 and 5, because 3×5 = 15. Some numbers have more than one factorization (way of being
factored). For instance, 12 can be factored as 1×12, 2×6, or 3×4. A number that can only be factored
as 1 times itself is called "prime". The first few primes are 2, 3, 5, 7, 11, and 13. The number 1 is not
regarded as a prime, and is usually not included in factorizations, because 1 goes in to everything. (The
number 1 is a bit boring in this context, so it gets ignored.)
You most often want to find the "prime factorization" of a number. That is, you usually want to
find the list of all the prime-number factors of a given number. The prime factorization does not include 1,
but does include every copy of every prime factor. For instance, the prime factorization of 8 is 2×2×2,
not just "2". Yes, 2 is the only factor, but you need three copies of it to multiply back to 8, so the prime
factorization includes all three copies.
On the other hand, the prime factorization includes ONLY the prime factors, not any products of
those factors. For instance, even though 2×2 = 4, and even though 4 is a divisor of 8, 4 is NOT in the
PRIME factorization of 8. That is because 8 does NOT equal 2×2×2×4! This accidental over-duplication
of factors is another reason why the prime factorization is often best: it avoids counting any factor too
many times. Suppose that you need to find the prime factorization of 24. Sometimes a student will just
list all the divisors of 24: 1, 2, 3, 4, 6, 8, 12, and 24. Then the student will make the product of all these
divisors: 1×2×3×4×6×8×12×24. But this equals 331776, not 24! So it's best to stick to the prime
factorization, even if the problem doesn't require it, in order to avoid either omitting a factor or else overduplicating one.
In the case of 24, you can find the prime factorization by taking 24 and dividing it by the smallest
whole number (other than 1) that goes into 24: 24 ÷ 2 = 12. Now divide out the smallest number that
goes into 12: 12 ÷ 2 = 6. Now divide out the smallest number that goes into 6: 6 ÷ 2 = 3. Then the
prime factorization is 2×2×2×3. An easy way of keeping track of the factorization is to do repeated
division. The steps in the repeated division look like this:
You’re
factoring
24
24
What’s the
smallest prime
that divides 24?
The smallest
prime is 2, so
divide by 2.
24
2 24
The
result is
12.
12
2 24
What is the
smallest prime
that divides 12?
It is 2, so divide
by 2.
The
result
is 6.
12
2 12
6
2 12
2 24
2 24
2 24
What is the
smallest prime
that divides 6?
It is 2 again, so
divide by 2.
The result
is 3, which
is prime,
so you’re
done.
6
2 6
3
2 6
2 12
2 12
2 12
2 24
2 24
2 24
Whenever you get to where the result (on top) of the division is a prime number, then you’ve gone
as far as you can, and your work is finished.
The nice thing about this repeated division is that, when you're done, the prime factorization is the
product of all the numbers around the outside. The factors are circled below (next page):
Copyright © Elizabeth Stapel 2002
All Rights Reserved
3
2 6
2 12
2 24
By the way, this repeated division is something that should probably be done on scratch-paper,
and not handed in as part of your homework.
Here's another example.
•
Find the prime factorization of 1050.
Do the repeated division:
What prime
divides 1050?
1050
Divide by 2;
get 525.
525
2 1050
Then the answer is:
Divide by 3;
get 175.
Divide by 5;
get 35.
Divide again by 5; the
result of 7 is prime.
Circle the prime
factors.
7
5 35
175
3 525
35
5 175
7
5 35
5 175
5 175
3 525
3 525
3 525
2 1050
2 1050
2 1050
2 1050
1050 = 2 × 3 × 5 × 5 × 7
Note: Some texts prefer that answers such as this be written using exponential notation, so that
2
the final answer would be written as 2×3×5 ×7.
Another example would be:
•
Find the prime factorization of 1092.
Do the repeated division:
13
7 91
3 273
2 546
2 1092
1092 = 2 × 2 × 3 × 7 × 13
2
This answer might also be written as 2
×3×7×13.
The above lesson is the print version of the “Factoring Numbers” web lesson on Purplemath, located at
http://www.purplemath.com/modules/factnumb.htm. The above text-based “Factoring Numbers” lesson was produced by
Elizabeth Stapel (stapel@purplemath.com), and may be distributed freely as long as this distribution is made at no charge and as
long as the copyright information is retained:
Copyright © Elizabeth Stapel 2002
All Rights Reserved