Download Number Theory

Number Theory
Yolanda McHenry, Ashley Courtney,
Tyler Williams, Jamiya Hagger
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.
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
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.
Factors and Multiples
Sieve of Eratosthenes- A systematic method for
Prime
Composite
Numbers
identifying
primeand
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
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
Divisibility Test
An aid in determining whether a natural number is divisible by
another natural number is called a divisibility test.
2
Number ends in 0,2,4,6,8.(The last
digit is even.)
9,489,994 ends in 4, it is divisible
by 2.
3
Sum of the digits is divisible by 3
897,432 is divisible by 3, since
8+9+7+4+3+2=33.
4
Last two digits form a number divisible
by 4.
7,693,432 is divisible by 4, since
32 is divisible by 4.
5
Number ends in 0 or 5.
8900 and 7635 are divisible by 5.
6
Number is divisible by both 2 and 3.
27,342 is divisible by 6 since it is
divisible by both 2 and 3.
8
Last three digits form a number
divisible by 8.
1,437,816 is divisible by 8, since
816 is divisible by 8.
9
Sum of the digits is divisible by 9.
428,376,105 is divisible by 9 since
the sum of digits is 36, which is
divisible by 9.
10
Last digit is 0
897,4663,940 is divisible by 10
12
Number is divisible by both 4 and 3.
376,984,032 is divisible by 12.
Divisibility Tests(cont’d)
123,216
2
Number ends in 0,2,4,6,or 8.( The last digit is even)
The last digit,8, is an even number therefore the 123,216 is
divisible by 2.
3 Sum of the digits is divisible by 3.
The sum of the digits,15, is divisible by 3 therefore
123,216 is divisible by 3.
4
Last two digits for a number divisible by 4.
The last two digits 16, are divisible by 4 therefore
123,216 is divisible by 4.
123,216
Divisibility Test (cont’d)
• The last digit does not end in a 5 or 0 therefore 123,216 is not
divisible by 5.
5 Number ends in 5 or 0
The number is divisible by both 2 and 3 therefore 123, 216 is divisible
by 6.
6
Number is divisible by both 2 and 3
• The last three digits 216, is divisible by 8 therefore 123,216 is
divisible by 8
8
Last three digits form a number divisible by 8
123,216
Divisibility
Tests (cont’d)
The sum of the digits ,15, is not divisible by 9
therefore 123,216 is not divisible by 9
9
Sum of the digits is divisible by 9
• The last digit does not end 0, therefore 123,216
is not divisible by 10.
10
The last digits is 0
• 123,216 is divisible by both 4 and 3 therefore it is
divisible by 12.
12
The number is divisible by both 4 and 3.