Factoring Numbers
"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 (more than
one 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 into 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: 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 do something like 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 over-duplicating one.
In the
case of
24,
you can find the prime factorization by taking
24
and dividing it by the smallest prime number that goes
into
24: 24 ÷ 2 = 12.
(Actually, the "smallest" part is not as important as
the "prime" part; the "smallest" part is mostly to make
your work easier, because dividing by smaller numbers is
simpler.) 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.
Since
3
is prime, you're done factoring, and the prime
factorization is
2×2×2×3.
An easy
way of keeping track of the factorization is to do
upside-down division. It looks like this:
24
The nice
thing about this upside-down division is that, when
you're done, the prime factorization is the product of
all the numbers around the outside. The factors are
circled in red above. By the way, this upside-down
division is something that should probably be done on
scratch-paper, and not handed in as part of your
homework.<
Some
texts prefer that answers such as this be written using
exponential
notation, in which case the final answer would be
written as
2×3×52×7.
You can do
the repeated division "right-side up", too, if you
prefer. The process works the same way, but the division
is reversed in orientation. The above problem would be
worked out like this:
I'll do
the repeated division:
1092 =
2 × 2 × 3 × 7 × 13
This answer
might also be written as
22×3×7×13.
By the way,
there are some divisibility rules that can help you find
the numbers to divide by. There are
many divisibility rules,
but the simplest to use are these:
- If
the number is even, then it's divisible by
2.
- If
the number's digits sum to a number that's divisible
by
3,
then the number itself is divisible by
3.
- If
the number ends with a
0
or a
5,
then it's divisible by
5.
Of course,
if the number is divisible twice by
2,
then it's divisible by
4;
if it's divisible by
2
and by
3,
then it's divisible by
6;
and if it's divisible twice by
3
(or if the sum of the digits is divisible by
9),
then it's divisible by
9.
But since you're finding the prime factorization, you
don't really care about these non-prime divisibility
rules. There is a rule for divisibility by
7,
but it's complicated enough that it's probably easier to
just do the division on your calculator and see if it
comes out even.
If you
run out of small primes and you're not done factoring,
then keep trying bigger and bigger primes (11,
13, 17, 19, 23,
etc) until you find something that works — or until you
reach primes whose squares are bigger than what you're
dividing into. Why? If your prime doesn't divide in,
then the only potential divisors are bigger primes.
Since the square of your prime is bigger than the
number, then a bigger prime must have as its remainder a
smaller number than your prime. The only smaller number
left, since all the smaller primes have been eliminated,
is
1.
So the number left must be prime, and you're done.
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