Thinking Mathematically Number Theory and the Real Number System 5.1 Prime and Composite Numbers

The Set of Natural Numbers

Divisibility If a and b are natural numbers, a is divisible by b if the operation of dividing a by b leaves a remainder of 0. This is the same as saying that b is a divisor of a, or b divides a. All three statements are symbolized by writing b|a. If b|a, then b is a factor of a

Rules of Divisibility Even numbers (last digit is even) are divisible by 2 Numbers ending in 0, 5 are divisible by 5 Numbers ending in 0 are divisible by 10 To be divisible by a composite number, must be divisible by factor of the composite number. Table 5.1

Example Divisibility Exercise Set 5.1 #5 2 3 4 5 6 8 9 10 12 Determine if 26,428 is divisible by each of the following numbers: 2 3 4 5 6 8 9 10 12

Prime Numbers Composite Numbers A prime number is a natural number greater than 1 that has only itself and 1 as factors. Composite Numbers A composite number is a natural number greater than 1 that is divisible by a number other than itself and 1.

The Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way (if the order of the factors is disregarded). Prime factorization The prime factors of a natural number can be found by constructing a “factor tree.” Write the given number as a product and continue to factor each composite number until only prime numbers remain.

Example: Prime Factorization Exercise Set 5.1 #33 Find the prime factorization of 663

Finding the Greatest Common Divisor of Two or More Numbers Using Prime Factorization To find the greatest common divisor of two or more numbers: Write the prime factorization of each number. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. [The GCD is the intersection of the two sets of factors]

Example: GCD Exercise Set 5.1 #49 Find the Greatest Common Divisor of 60 and 108

Finding the Least Common Multiple Using Prime Factorization To find the least common multiple of two or more numbers: Write the prime factorization of each number. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. Form the product of the numbers from step 2. The least common multiple is the product of these factors. [The LCM is the union of the two sets of factors]

Example: LCM Exercise Set 5.1 #63 Find the Least Common Multiple of 72 and 120

Thinking Mathematically Number Theory and the Real Number System 5.1 Prime and Composite Numbers