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 bΧa. 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 Sieve of Eratosthenes- A systematic method for identifying prime numbers in a list of numbers. Step 1: Circle the prime number 2, then cross out all other multiples of 2 Step 2: Circle three, then cross out all other multiples of 3. Step 4: Continue process for all other primes less than or equal to square root of the last number. Step 5: Circle all remaining numbers that are not crossed out. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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 3Sum of the digits is divisible by 3. The sum of the digits,15, is divisible by 3 therefore 123,216 is divisible by 3. 4Last two digits for a number divisible by 4. The last two digits 16, are divisible by 4 therefore 123,216 is 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 5Number ends in 5 or 0 6Number is divisible by both 2 and 3 8Last 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 10The last digits is 0 12The number is divisible by both 4 and 3.