5.4 Prime and Composite Numbers

Math 365 Lecture Notes © S. Nite 3/2/2012 Page 1 of 3 Section 5-4

One method to determine factors of a natural number is to use squares of paper or cubes to represent the number as a rectangle.

Example : Find the positive factors of 12.

A positive integer with exactly two distinct, positive divisors is a prime number .

A positive integer greater than 1 that has a positive factor other than 1 and itself is a composite number .

Prime Factorization

In the grade 7 Focal Points, “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” (p. 19).

An expression of a number as a product of factors is a factorization . A factorization containing only prime numbers is a prime factorization .

Theorem 5-23 Fundamental Theorem of Arithmetic 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.

Note : The primes in the prime factorization of a number are typically listed in increasing order from left to right. Exponential notation may be used.

Math 365 Lecture Notes © S. Nite 3/2/2012 Page 2 of 3 Section 5-4

Number of Divisors

Example : Find the number of divisors for 24.

The prime factorization can be used with the fundamental counting principle to find the number of positive divisors.

Theorem 5-24 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 n1 n2 nk p1 ⋅ p1 ⋅...⋅ pk as ( n1 + 1)( n2 + 1) …(nk + 1) positive divisors.

Example : Find the number of divisors for 16,200.

Determine Whether a Number Is Prime

Theorem 5-25 If d is a divisor of n, then n is also a divisor of n. d

Theorem 5-26 If n is composite, then n has a prime factor p such that p2 ≤ n.

Theorem 5-27 If n is an integer greater than 1 and not divisible by any prime p, such that p2 ≤ n, then n is prime.

One way to find all the primes less than a given number is to use the Sieve of Eratosthenes .

1 11 21 31 41 51 61 71 81 91 2 12 22 32 42 52 62 72 82 92 3 13 23 33 43 53 63 73 83 93 4 14 24 34 44 54 64 74 84 94 5 15 25 35 45 55 65 75 85 95 6 16 26 36 46 56 66 76 86 96 7 17 27 37 47 57 67 77 87 97 8 18 28 38 48 58 68 78 88 98 9 19 29 39 49 59 69 79 89 99 10 20 30 40 50 60 70 80 90 100

More About Primes

There is a whole branch of research and study of primes.