Transcript Prime and Composite Factor Trees

Prime Numbers
and Prime
Factorization
Factors
• Factors are the numbers you multiply
together to get a product.
• For example, the product 24 has several
factors.
• 24 = 1 x 24
• 24 = 2 x 12
• 24 = 3 x 8
• 24 = 4 x 6
• SO, the factors are 1, 2, 3, 4, 6, 8, 12, 24
Finding Factors
• Start with 1 times the number.
• Try 2, 3, 4, etc.
• When you repeat your factors, cross out
the repeat - you’re done at this point.
• If you get doubles (such as 4 x 4), then
you’re done. Repeats or doubles let
you know you’re done.
What are the factors of 16?
1 x 16
2x8
3 x ??
4x4
3 is not a factor, so cross it out
doubles = done
The factors of 16 are
1,2,4,8,16
What are the factors of 18?
1
2
3
4
5
6
x
x
x
x
x
x
18
9
6
??
??
3
The factors are
1,2,3,6,9,18
Repeat! Cross it out!
We’re done!
What are the factors of 7?
1x7
2 x ??
3 x ??
4
5
6
7
x
x
x
x
??
??
??
1
The only factors
of 7 are 1,7
This works, but it is a repeat. We
are done.
Prime and Composite
Numbers
Prime numbers are
numbers that only have
two factors: one, and the
number itself.
EXAMPLES:
3, 5, 7, 11, 31
Composite numbers
have more than two
factors.
EXAMPLES:
6, 15, 18, 30, 100
A Product of Primes
• Every composite number can be
expressed as a product of prime
numbers.
• This is called prime factorization.
Example
•15 is a composite
number.
•It can be expressed as
a product of primes:
3x5
To find the prime factorization:
1. Divide the number by the first prime
number possible.
2. Circle the prime number, and continue
with the other factor.
3. Divide the new factor by a prime
number.
4. Continue this process until the only
numbers you have left are prime
numbers.
Remember the Prime Number
List:
• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29,
31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97…
Example: Prime Factorization of 100.
100
2 is a prime
number, so we are
done with it.
100 ÷ 2 = 50. Two is
the first prime number
that goes into 100.
2 X 50
Now we deal with the
50. Divide it by 2 to get
the next factors.
2 X 25
Both numbers are prime,
leaving us with all primes.
25 is not divisible by
the first prime, 2. The
next prime, 3, does not
work either. We must
divide by 5 to get a
factor.
5x5
What’s the Answer?
• Now, we just list our factors with
multiplication signs between them. Use
the circled prime numbers.
• 2x2x5x5
• We have listed 100 as a product of
prime numbers.
Exponent Form
• We have just listed our prime factorization for
100 as being 2 x 2 x 5 x 5. This is repeated
multiplication. Repeated multiplication can be
expressed with exponents.
• Our prime numbers are our bases. The
number of times the prime number is written
is the exponent.
2
• 2 x 2 can be expressed in exponent form: 2
2
• 5 x 5 can be expressed as 5
• Put it together, and 2 x 2 x 5 x 5 is more
simply put as
2
2
2 x5
Another Example
420
2 x 210
2 x 105
2
2 x3x5x7
3 x 35
5x7
Try this on your own:
54
Answer:
2x3
3
Homework Time!