5-4 Prime and Composite Numbers
Prime and Composite Numbers
Prime Factorization
Number of Divisors
Determining if a Number is Prime
More About Primes
Prime and Composite Numbers
Students should recognize that different types of numbers have particular
characteristics; for example,
ample, square numbers have an odd number of factors and
prime numbers have only two factors.
The following rectangles represent the number 18.
The number 18 has 6 positive divisors: 1, 2, 3, 6, 9 and 18.
Number of Positive Divisors
Prime and Composite Numbers
Below each number listed
sted across the top,
we identify numbers less than or equal to
37 that have that number of positive
divisors.
Prime number
Any positive integer with exactly two distinct, positive divisors
Composite number
Any integer greater than 1 that has a positive factor other than 1 and itself
Example 1 Show that the following numbers are composite.
a. 1564
b. 2781
c. 1001
d. 3 · 5 · 7 · 11 · 13 + 1
Prime Factorization
Students continue to develop their understanding of multiplication and division
and the structure of numbers by determining if a counting number greater than 1
is a prime, and if it is not, by factoring it into a product of primes.
Composite numbers can be expressed as products of two or more whole numbers
greater than 1.
Each expression of a number as a product of factors is a factorization.
A factorization containing only prime numbers is a prime factorization.
Fundamental Theorem of Arithmetic (Unique Factorization Theorem)
Each composite number can be written as a product of primes in one and only
one way except for the order of the prime factors in the product.
Prime Factorization
To find the prime factorization of a composite number, rewrite the number as a
product of two smaller natural numbers. If these smaller numbers are both
prime, you are finished. If either is not prime, then rewrite it as the product of
smaller natural numbers. Continue until all the factors are prime.
The two trees produce the same prime factorization,
factorization, except for the order in
which the primes appear in the products.
Prime Factorization
We can also determine the prime factorization by dividing with the least prime, 2,
if possible. If not, we try the next larger prime as a divisor. Once we find
f a prime
that divides the number, we continue by finding smallest prime that divides that
quotient, etc.
Number of Divisors
How many positive divisors does 24 have? We are not asking how many prime
divisors, just the number of divisors – any divisors.
Since 1 is a divisor of 24, then 24/1 = 24 is a divisor of 24.
Since 2 is a divisor of 24, then 24/2 = 12 is a divisor of 24.
Since 3 is a divisor of 24, then 24/3 = 8 is a divisor of 24.
Since 4 is a divisor of 24, then 24/4 = 6 is a divisor of 24.
her way to think of the number of positive divisors of 24 is to consider the
Another
prime factorization
23 = 8 has four divisors. 3 has two divisors.
Using the Fundamental Counting Principle, there are 4 × 2 = 8 divisors of 24.
If p and q are different primes, then pnqm has (n + 1)(m + 1) positive divisors.
In general, if p1, p2, …, pk are primes, and n1, n2, …, nk are whole numbers, then
has
positive divisors.
Example 2 a. Find the number of positive divisors of 1,000,000.
b. Find the number of positive divisors of 21010.
Determining if a Number is Prime
To determine if a number is prime, you must check only divisibility by prime
numbers less than the given number.
For example, to determine if 97 is prime, we must try dividing 97 by the prime
numbers: 2, 3, 5, and so on as long as the prime is less than 97.
97
If none of these prime numbers divide 97, then 97 is prime.
Upon checking, we determine that 2, 3, 5, 7 do not divide 97.
Assume that p is a prime greater than 7 and p | 97. Then 97/p also divides 97.
Because p ≥ 11, then 97/p
p must be less than 10 and hence cannot divide 97.
If d is a divisor of n, then
is also a divisor of n.
If n is composite, then n has a prime factor p such that p2 ≤ n.
If n is an integer greater than 1 and not divisible by any prime p,
p such that p2 ≤ n,
then n is prime.
Note: Because p2 ≤ n implies that
than or equal to
it is enough to check if any prime less
is a divisor of n.
Example 3 a. Is 397 composite or prime?
b. Is 91 composite or prime?
Sieve of Eratosthenes
One way to find all the primes less than a given number is to use the Sieve of
Eratosthenes.
If all the natural numbers greater than
1 are considered (or placed in the
sieve), the numbers that are not prime
are methodically crossed out (or drop
through the holes of the sieve). The
remaining numbers are prime.
5.4 Homework # A-2,
2, 3, 4, 5, 6, 7, 9, 10, 13, 14, 18, 19