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