Download prime number - Dr. Ben Weng

Discrete Mathematics
Properties of Integers
Pangyen Weng, Ph.D
Metropolitan State University
Prime Factorization
and
Greatest Common Divisors
Prime and Composite Numbers
(In this lesson all numbers discussed are positive.)
A prime number is an integer that cannot be factored into the
product of two smaller integers.
Prime and Composite Numbers
(In this lesson all numbers discussed are positive.)
A prime number is an integer that cannot be factored into the
product of two smaller integers.
A composite number is an integer that can be factored into the
product of two smaller integers.
Prime and Composite Numbers
(In this lesson all numbers discussed are positive.)
A prime number is an integer that cannot be factored into the
product of two smaller integers.
A composite number is an integer that can be factored into the
product of two smaller integers.
– We can factor a composite number all the way down to the product
of some prime numbers.
Prime Factorization
When factoring a number, if we…
– factor down to prime numbers
– arrange the prime factors in increasing order and
– use exponents when necessary,
then we get a prime factorization for a number.
Prime Factorization
When factoring a number, if we…
– factor down to prime numbers
– arrange the prime factors in increasing order and
– use exponents when necessary,
then we get a prime factorization for a number.
Example. Find the prime factorization for 180.
Prime Factorization
When factoring a number, if we…
– factor down to prime numbers
– arrange the prime factors in increasing order and
– use exponents when necessary,
then we get a prime factorization for a number.
Example. Find the prime factorization for 180.
Exercise. How about 4004?
Common Divisors
If integer 𝑎 is a common divisor (or factor) of 𝑚 and 𝑛 then
𝑚 = 𝑎 ⋅ 𝑘 and 𝑛 = 𝑎 ⋅ ℎ for some integers 𝑘 and ℎ.
Common Divisors
If integer 𝑎 is a common divisor (or factor) of 𝑚 and 𝑛 then
𝑚 = 𝑎 ⋅ 𝑘 and 𝑛 = 𝑎 ⋅ ℎ for some integers 𝑘 and ℎ.
Hint. Can the factored forms of 𝑚 and 𝑛 help us find 𝑎?
Common Divisors
If integer 𝑎 is a common divisor (or factor) of 𝑚 and 𝑛 then
𝑚 = 𝑎 ⋅ 𝑘 and 𝑛 = 𝑎 ⋅ ℎ for some integers 𝑘 and ℎ.
Hint. Can the factored forms of 𝑚 and 𝑛 help us find 𝑎?
Example. Find a few common divisors of 54 and 36.
Common Divisors
If integer 𝑎 is a common divisor (or factor) of 𝑚 and 𝑛 then
𝑚 = 𝑎 ⋅ 𝑘 and 𝑛 = 𝑎 ⋅ ℎ for some integers 𝑘 and ℎ.
Hint. Can the factored forms of 𝑚 and 𝑛 help us find 𝑎?
Example. Find a few common divisors of 54 and 36.
Exercise. Find a few common divisors of
𝑚 = 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 7 ⋅ 7,
𝑛 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 7 ⋅ 11
Greatest Common Divisors
Example. Find the greatest common divisor of 54 and 36.
Greatest Common Divisors
Example. Find the greatest common divisor of 54 and 36.
Do you observe any strategy?
Greatest Common Divisors
Example. Find the greatest common divisor of 54 and 36.
Do you observe any strategy?
Exercise. Find gcd(𝑚, 𝑛) for
𝑚 = 22 ⋅ 3 ⋅ 5 ⋅ 73 ,
𝑛 = 23 ⋅ 32 ⋅ 7 ⋅ 11