Survey of Math - Section 5.1: Number Theory
conjecture
A supposition or hypothesis that has not been proved or disproved is known as
Fermat number
Fermat conjectured that each number of the form 2(2n+1), now referred to as a ___, was prime for each natural number n.
2N-1
Mersenne primes form
Mersenne prime numbers. *Note: Marin Mersenne, a seventeenth-century monk, found that numbers of the form 2n-1, are often prime numbers when n is a prime number. Numbers of the form 2n-1, that are prime are referred to as Mersenne primes. The first 10 Mersenne primes occur when nequals2, 3, 5, 7, 13, 17, 19, 31, 61, 89.
Prime numbers of the form 2n-1, where n is a prime number, are known as ___
28 is a multiple of 7.
Review!!! Determine whether the following statement is true or false. Modify each false statement to make it a true statement. 7 is a multiple of 28.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
Review!!! Prime numbers up to 100
5 , prime
The first Fermat number is ___ and it is ___.
Christian Goldbach
Who conjectured that every even number greater than or equal to 4 can be represented as the sum of two (not necessarily distinct) prime numbers.
Prime numbers
is a natural number greater than 1 that has exactly two factors (or divisors), itself and 1. Example: 17 = 1x17
Composite *Note: A natural number that is divisible by a number other than itself and 1 is known as a composite number.
A natural number that is divisible by a number other than itself and 1 is known as a ___ number.
Perfect number
A number whose proper factors (factors other than the number itself) add up to the number is called a perfect number.___
Prime *Note: A natural number greater than 1 that has only itself and 1 as factors is called a prime number. The first several prime numbers are 2, 3, 5, 7, 11, 13, and 17.
A natural number greater than 1 that has only itself and 1 as factors is called a/an __ number.
0 *Note: When a is divisible by b it means that b divides a evenly. When this is the case, the quotient has a remainder of 0. Therefore, if a is divisible by b, then a divided by b has a remainder of 0.
If a is divisible by b, then a divided by b has a remainder of
Mersenne primes
Numbers of the form 2N-1 that are prime are referred to as _____.
True
Review!!! Determine whether the following statement is true or false. Modify each false statement to make it a true statement. 9 is a factor of 63.
Every composite number can be expressed as a UNIQUE product of prime numbers.
Review!!! The Fundamental Theorem of Arithmetic
Determine whether 963,888 is divisible by 2, 3, and 4. A) 2 - YES, because the last digit 8 is an even number. B) 3 - YES, because 9+6+3+8+8+8= "42" is divisible by 3. C) 4 - YES, because the last TWO digits "88" is divisible by 4. D) 5 - NO, because the last digit is not a 5 or a 0. E) 6 - YES, because it is divisible by 2 and 3. F) 8 - YES, because the last THREE digits "888" is divisible by 8. G) 9 - NO, because the 9+6+3+8+8+8= "42" is not divisible by 9. H) 10 - NO, because the last digit is not 0.
Review!!! Using the divisibility Rules Examples
Greatest Common Divisor (GCD)
The ___ of a set of natural numbers is the largest natural number that divides (without remainder) even number in that set.
Least Common Multiple (LCM)
The ___ of a set of natural numbers is the smallest natural number that is divisible (without remainder) by each element of the set.
Greatest Common Denominator
The largest natural number that divides (without remainder) each number in a set of numbers is known as the ___ of the set of numbers.
Factors
The natural numbers that are multiplied together are called ___ of the product.
257 , prime
The third Fermat number is ___ and it is ___.
2, 3, 5, 7, 13, 17, 19, 31, 61, 89.
Mersenne primes occurs when N =
10 use GCD (keyword = even/same)
Review!!! Elizabeth is a manager at a nursery and is in charge of displaying potted trees in rows. Elizabeth has 250 citrus trees and 120 palm trees. She wants to make rows of trees so that each row has the same number of trees and each tree is in a row. If the citrus trees and the palm trees must not be mixed in the rows, what is the largest number of trees that she can have in a row?
80 Use GCD (keyword = largest number of cars)
Review!!! Jen collects toy sports cars. She has 400 orange cars and 160 blue cars. She wants to line up the cars in groups so that each group has the same number of cars and contains only orange cars or only blue cars. What is the largest number of cars that she can have in a group?
A, B, C, F, G, H, K, L, M, N, Q, R, T, U
Review!!! Jerrett is setting up chairs for his school band concert. He needs to put 945 chairs on the gymnasium floor in rows of equal size, and there must be at least 3 chairs in each row and at least 3 rows. List the number of rows and the number of chairs in each row that are possible. Select all that apply. A. 35 rows by 27 chairs B. 189 rows by 5 chairs C. 5 rows by 189 chairs D. 3 rows by 260 chairs E. 1 row by 945 chairs F. 63 rows by 15 chairs G. 135 rows by 7 chairs H. 3 rows by 315 chairs I. 249 rows by 4 chairs J. 259 rows by 3 chairs K. 105 rows by 9 chairs L. 9 rows by 105 chairs M. 15 rows by 63 chairs N. 27 rows by 35 chairs O. 260 rows by 3 chairs P. 945 rows by 1 chair Q. 45 rows by 21 chairs R. 21 rows by 45 chairs S. 4 rows by 249 chairs T. 7 rows by 135 chairs U. 315 rows by 3 chairs
April 19
Review!!! You and your brother both work the 4:00 P.M. to midnight shift. You have every ninth night off. Your brother has every sixth night off. Both of you were off on April 1. Your brother would like to see a movie with you. When will the two of you have the same night off again?
Another method that can be used to find the greatest common divisor is known as the Euclidean algorithm. This process is illustrated by finding the GCD of 60 and 220 below. First divide 220 by 60 as shown below. Disregard the quotient 3 and then divide 60 by the remainder 40. Continue this process of dividing the divisors by the remainders until a remainder of 0 is obtained. The divisor in the last division, in which the remainder is 0, is the GCD. 220/60 = 3 with remainder 40 60/40 = 1 with remainder 20 40/40 = 2 with remainder 0 Since 40 divided by 20 had a remainder of 0, the GCD is 20. Use the Euclidean algorithm to find the GCD. 160, 200 What is the GCD of 160 and 200? Answer: 40
Review!!! Euclidean Algorithm
Least Common Multiple
The smallest natural number that is divisible (without remainder) by each number in a set of numbers is known as the least common ___ of the set of numbers.
120 Note: To find how many days it will be before she cuts the grass and trims her shrubs on the same day, find the least common multiple of 10 and 24. First determine the prime factorization of each number. Then list each prime factor with the greatest exponent that appears in any of the prime factorizations. Finally determine the product of the factors. keyword = same day / cut
Review!!! Example on LCM Brenda cuts her grass every 10 days and trims her shrubs every 24 days. If Brenda cut her grass and trimmed her shrubs on June 1, how many days will it be before she cuts her grass and trims her shrubs on the same day again?
1) Determine the prime factorization of each number. 2) List each prime factor with the SMALLEST exponent that appears in each of the prime factorization. 3) Determine the product of the factors found in Step 2.
Review!!! Procedure to determine the GCD of two or more numbers:
1) Determine the prime factorization of each number. 2) List each prime factor with the GREATEST exponent that appears in any of the prime factorization. 3) Determine the product of the factors found in Step 2.
Review!!! Procedure to determine the LCM of two or more numbers.
(2) = the number is even. Example: 924 is divisible by 2, since 924 is even. (3) = the sum of the digits of the number is divisible by 3. Example: 924 is divisible by 3, since the sum of the digits is 9+2+4=15, and 15 is divisible by 3. (4) = the number formed by the last two digits of the number is divisible by 4. Example: 924 is divisible by 4, since the number formed by the last two digits, 24, is divisible by 4. (5) = the number of ends in 0 or 5. Example: 265 is divisible by 5, since the number ends in 5. (6) = the number is divisible by both 2 and 3. Example: 924 is divisible by 6, since it is divisible by both 2 and 3. (8) = the number formed by the last three digits of the number is divisible by 8. Example: 5824 is divisible by 8, since the number formed by the last three digits, 824, is divisible by 8. (9) = the sum of the digits of the number is divisible by 9. Example: 837 is divisible by 9, since the sum of the digits, 8+3+7, =18, is divisible by 9. (10) = the number ends in 0 Example: 290 is divisible by 10, since the number ends in 0.
Review!!! Rules of divisibility
Greatest Common Divisor (GCD) *Note: The largest natural number that divides (without remainder) each number in a set of numbers is known as the greatest common divisor of the set of numbers. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12. The divisors of 20 are 1, 2, 4, 5, 10, and 20. The greatest common divisor of 12 and 20 is 4 because 4 is the largest natural number that is a divisor of both.
The largest natural number that divides (without remainder) each number in a set of numbers is known as the ___ of the set of numbers.
Counting numbers or natural numbers
The numbers we use to count are called....
No. Any other consecutive pair of natural numbers will include an even number. All even numbers are divisible by 2, so this number would be composite.
The primes 2 and 3 are consecutive natural numbers. Is there another pair of consecutive natural numbers both of which are prime? Explain.
Prime factorization
The process of breaking a given composite number down into a product of prime numbers is called ___
17 , prime
The second Fermat number is ___ and it is ___.
Multiple *Note: The smallest natural number that is divisible (without remainder) by each number in a set of numbers is known as the least common multiple of the set of numbers.
The smallest natural number that is divisible (without remainder) by each number in a set of numbers is known as the least common ___ of the set of numbers.
number theory
The study of numbers and their properties is known as
Twin primes
are primes of the form p and p + 2 Example: 3 and 5, and 5 and 7, and 11 and 13
Composite number
is a natural number that is divisible by a number other than itself and 1. Example: 6 can also be written as 2x6
