1 / 13

Thinking Mathematically

Number Theory and the Real Number System 5.1 Prime and Composite Numbers. Thinking Mathematically. The Set of Natural Numbers. N = {1,2,3,4,5,6,7, 8, 9, 10, 11, ... }. Divisibility.

axel-copeland
Download Presentation

Thinking Mathematically

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  2. The Set of Natural Numbers N = {1,2,3,4,5,6,7, 8, 9, 10, 11, ... }.

  3. 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 bdividesa. All three statements are symbolized by writing b|a. If b|a, then b is a factor of a

  4. 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

  5. Example Divisibility Exercise Set 5.1 #5 Determine if 26,428 is divisible by each of the following numbers:

  6. Prime 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.

  7. 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.

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

  9. 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]

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

  11. 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]

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

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

More Related