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# Number Theory

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• ### Number Theory

1. 1. Number Theory Yolanda McHenry, Ashley Courtney, Tyler Williams, Jamiya Hagger
2. 2. Natural Numbers <ul><ul><ul><li>The set of natural numbers is also called the set of counting numbers or positive numbers. </li></ul></ul></ul><ul><ul><ul><li>{1,2,3,4…} </li></ul></ul></ul><ul><ul><ul><li>Number theory deals with the study of the properties of this set of numbers ({1,2,3,4…})and the </li></ul></ul></ul><ul><ul><ul><li>key concept to number theory is divisibility. </li></ul></ul></ul>
3. 3. Divisibility <ul><li>One counting number is divisible by another if the operation of dividing the first by the second leaves a remainder of 0 . </li></ul><ul><li>Divisibility - The natural number a is divisible by the natural number b if there exists a natural number k such that a=bk . </li></ul><ul><li>45=9k; k is 5 </li></ul><ul><li>If b divides a, then we write b|a. </li></ul><ul><li>If b does not divide a, then we write b Χ a. </li></ul><ul><li>9|45 </li></ul>
4. 4. Factors and Multiples <ul><li>If the natural number a is divisible by the natural number b, then b is a divisor or factor of a. </li></ul><ul><li>20=10k </li></ul><ul><li>All factors of b are 1,2,4,5,10,20. </li></ul><ul><li>What are all the factors of 15? </li></ul><ul><li>1,3,5, and 15 </li></ul><ul><li>If the natural number a is divisible by the natural number b, then a is a multiple of b. </li></ul><ul><li>Other multiples of b include 30,40,50,60, 70 and so on. </li></ul><ul><li>What are some multiples of 5? </li></ul><ul><li>10,15,20, 25 and so on. </li></ul>
5. 5. Prime and Composite Numbers
6. 6. Prime and Composite Numbers <ul><li>Prime Number-A natural number greater than 1 that has only itself and 1 as factors. </li></ul><ul><li>2,3,5,7 and 11 are the first five prime numbers </li></ul><ul><li>Composite Number-A natural number greater than 1 that is not prime is called composite. </li></ul><ul><li>4,6,8,9, and 10 are the first five composite numbers </li></ul>
7. 7. Divisibility Test <ul><li>An aid in determining whether a natural number is divisible by another natural number is called a divisibility test. </li></ul>
8. 9. Divisibility Tests(cont’d) The last digit,8, is an even number therefore the 123,216 is divisible by 2. 123,216 The sum of the digits,15, is divisible by 3 therefore 123,216 is divisible by 3. The last two digits 16, are divisible by 4 therefore 123,216 is divisible by 4. 3 Sum of the digits is divisible by 3. 4 Last two digits for a number divisible by 4.
9. 10. Divisibility Test (cont’d) <ul><li>123,216 </li></ul><ul><li>The last digit does not end in a 5 or 0 therefore 123,216 is not divisible by 5. </li></ul><ul><li>The number is divisible by both 2 and 3 therefore 123, 216 is divisible by 6. </li></ul><ul><li>The last three digits 216, is divisible by 8 therefore 123,216 is divisible by 8 </li></ul>5 Number ends in 5 or 0 6 Number is divisible by both 2 and 3 8 Last three digits form a number divisible by 8
10. 11. Divisibility Tests (cont’d) <ul><li>123,216 </li></ul><ul><li>The sum of the digits ,15, is not divisible by 9 therefore 123,216 is not divisible by 9 </li></ul><ul><li>The last digit does not end 0, therefore 123,216 is not divisible by 10. </li></ul><ul><li>123,216 is divisible by both 4 and 3 therefore it is divisible by 12. </li></ul>9 Sum of the digits is divisible by 9 10 The last digits is 0 12 The number is divisible by both 4 and 3.