Transcript Number Theory

So, what are you learning in math?
We’re trying to find Least
Common Multiple and Greatest
Common Factor.
They must be really lost…people have
been looking for them since I was a child.
• A prime number is a natural number that has
exactly two positive factors, 1 and itself.
• A composite number is a natural number that
has more than two positive factors.
•
(Now might be a good time to review divisibility rules.)
• 0 is neither prime nor composite because it is
not a natural number.
• 1 is neither prime nor composite because it only
has one factor, 1.
Hut two! Hut three! Hut Five! Seven! Eleven! 13! 17! 19!
23! 29! 31! 37! 41! 43! 47! 53! 59! 61! 67! .... And Deion
Sanders thought HE was Prime Time!
• The Fundamental Theorem of Arithmetic
states that every composite number can
be written as the product of prime
numbers in exactly one way if you ignore
the order of the factors.
See
video
• Prime factorization is a number written as a
product of all its prime factors.
• You can use factor trees to find the prime
factorization of a number.
– You might have different branches than someone
else, but you should end up with the same prime
numbers on the bottom of all branches.
– Let’s look at two ways to find the prime factorization of
the number 12.
12 = 2 x 2 x 3 = 22 x 3
12
6
3
2
2
12
3
3
4
2
2
See
video
• The greatest common factor is the largest factor that two or
more numbers have in common.
• You might remember the listing method to find the
greatest common factor, GCF.
factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
factors of 54: 1, 2, 3, 6, 9, 18, 27, 54
GCF(36,54) = 18
• You can also use prime factorization. (GCF is the product
of all prime factors the numbers have in common)
36 = 2 x 2 x 3 x 3
54 = 2 x 3 x 3 x 3
GCF(36,54) = 2 x 3 x 3 = 18
See
video
• The least common multiple is the smallest natural number
that is a common multiple of two or more numbers .
• You might remember the listing method to find the least
common multiple, LCM.
0 is a multiple of
multiples of 36: 0 , 36, 72, 108 , 144
, 180but
, …LCM
each,
needs
be >
0. , …
multiples of 26: 0 , 26, 52 , 78, 104
, 130to
, 156
, 182
…but it’s not always the easiest method.
You can also use prime factorization. (Multiply all prime factors but
don’t include factors you already listed from a different number.)
26 = 2 x 13
36 = 2 x 2 x 3 x 3
LCM(26,36) = 2 x 13 x 2 x 3 x 3 = 26 x 18 = 468
Practice
• Here is a great website to practice making
factor trees. If you ask for two trees, you
will be asked to find GCF and LCM after
finding the prime factors. You can even
select use to pick the numbers yourself.
• Enjoy!