MATA02H3 Lecture 7: Prime/Composite Numbers & Eratosthenes Sieve

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MATA02H3 Lecture Notes - Lecture 6: Prime Number, Euclidean Algorithm, Linear Combination

MATA02H3 Lecture 7: Prime/Composite Numbers & Eratosthenes Sieve

MATA02H3 Lecture 8: Composite Consecutive Natural Numbers, Twin Primes, Positive Divisors and Euler (Φ) Function

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Mata02 - lecture 7 - prime/composite numbers & eratosthenes sieve. A prime number is a natural number greater than 1, that are only divisible by 1 and itself. A composite number is a natural number that is divisible by more numbers than just 1 and itself. 1 is neither a prime number, or a composite number. Ans: there are infinitely many prime numbers (euclid proved it refer to lecture 6) Determining if a number is prime or not. To figure out if a number is a prime number or not, divide the number by prime numbers. Theorem: if a number is composite (let"s call this number n"), then it must be divisible by a prime less than or equal to n. Proof: if n" is composite, then it"s a non-prime number that is greater than 1, and is divisible by multiple prime numbers. Since n" is composite, then n=a b where a,b are positive integers other than 1.

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In this exercise you will use an algorithm, called the sieve of Eratosthenes,
that isolates prime numbers. (Eratosthenes lived from about 276 to 194
b.c. He was also known for his computation of the circumference of the
earth and for his treatises on the theater, moral philosophy, and history.)
Here is how it works. List the integers from 2 through n. Circle the
number 2, then cross off all the higher multiples of 2 (namely, 4, 6, 8,
10, etc.). Circle 3, which is the first remaining number after 2. Then
cross off all the multiples of 3 that have not yet been crossed off. Then
circle 5. Et cetera. When you are finished with the repeated sieving
operations, only the primes will remain on the list, and they will have been
circled for emphasis. The exercise: perform the sieve of Eratosthenes with
n = 120. As you do this, cross off the multiples of different primes by using
different kinds of slashes (slashing in different directions, or using different
colors), and provide a key for the reader. By this device the reader will
immediately know the smallest prime divisor of each composite number
on your list. (In view of Exercise 4, after the multiples of 7 have been
crossed off, the remaining numbers will all be prime and can be circled to
mark this fact.)

General Math Terms A product B. digit C. numeral D. compass E. equation F. formula G. negative H. sum I. mean J. statistics K. positive L. percent M. straight-edge N. squareroot O. quotient P. prime numbers Q. difference R. addend S. minuend T. divisor Greater than zero: plus. A number being added to another number. A mathematical sequence whose verb is equal (=). The result of addition. The number doing the dividing in a division problem. An instrument used to construct straight lines. A mathematical expression of a natural law. The result of multiplication. The number from which another number is being subtracted. Any non-negative numeral less than ten. Less than zero: minus. An instrument used to construct circles and arcs of circles. A number that produces another number (it's square) when multiplied by itself. The result of subtraction. A symbol or name of a number. The average: the sum of the elements divided by the number of elements. The result of division. Parts per hundred: hundredths. Natural numbers greater than one that have no other factors except one and themselves. Set of numerical data. Geometry A. chord B. diameter C. radius D. perimeter E. cylinder F. surface area G. right angle H. square I. Hypotenuse J. obtuse angle K. vertex L point M. ray N. bisect O arc P. hexagon Q. pentagon R. polygon S. perpendicular T. rectangle U. rhombus V. triangle W. circumference X cube Y pyramid Z. sphere A six-sided polygon. The common end point of the sides of an angle. An angle measuring more than 90 degree. A long, round object with flat, circular ends. An angle of 90 degree. The side of a right triangle opposite the right angle. The total area of all the surfaces of an object. To multiply a number by itself. A location, the place where two lines cross or intersect. A closed geometric figure made up of three or more line segments. The perimeter of a circle: the distance around a circle. A shape having the same dimensions of length, height, and width. A half-line with a designated end point which extends in only one direction without stopping. A figure having a polygon base and triangular sides which meet in a point at the top. Divide an angle into two equal angles. A part of a circle

1. Evaluate the expression 2x - 2 for the value of x = -3

2. Factor: xsquare + 4x - 32 (x-4)(x+8) (x-4)(x-8) (x-4)(x-8) (x-16)

3. Factor: xsquare - 25 (x-5) (x+5)2 (x+5)(x-5) (x-5)(x-5)

4. Evaluate -u2v3 when u=4 and v=-2 128 64 32 -128

5. Using an online calculator, determine whether the value of x=7 is a solution of the equation. x Yes No

6. Explain why x=7 IS or IS NOT a solution of the equation. 7 + 1/x+2=8

7. Solve the equation and check your solution: -2 - 4x = 30

8. Solve the equation and check your solution: 67x - 24 = 3x + 8(8x - 3) 67 -3 -67 All real numbers

9. Solve the equation and check your solution. 5x/2 - x/6 = 14 10 6 7 9

10. Solve the quadratic equation by factoring: xsquare - 6x + 5 = 0 -1, 5 -1,-5 1,-5 1,5

11. The imaginary number i represents:

12. Find real numbers a and b such that the equation is true: a+ bi = 14 + 2i a=16, b=4 a=18, b=6 a=14, b=2 a=15, b=14

13. Find all solutions to the following equation. /4x - 8 = /4x + 9 x = -17/4 x=9 No solution x=-17

14. In a given amount of time, James drove twice as far as Rachel. Altogether they drove 120 miles. Write an equation to help find the number of miles driven by each. Use R as the variable to represent miles driven by Rachel and J for Jamesâ miles.

15. Karen works for $10 an hour. A total of 25% of her salary is deducted for taxes and insurance. She is trying to save $450 for a new set of tires. Write an equation to help determine how many hours she must work to take home $450 if she saves all of her earnings. Be sure to save the equation you write, as you will use it to answer the next question in this quiz. 10h - .25=450 10h + .25(10h)=450 h -.25 (10h)=450 10h - .25(10h)=450

16. Using the equation you wrote in the previous problem, find the actual amount of hours Karen must work to take home $450 if she saves all of her earnings. Show your work. 17. The length of a rectangular map is 15 inches and the perimeter is 50 inches. Find the width. Hint: Perimeter = width + length + width + length

18. Twice a number is added to the number and the answer is 80. Write an equation to solve this problem. 2n + n=80 2n = 80 2n + 80=n 80 + n=2n

19. If 4 is subtracted from twice a number, the result is 10 less than the number. Write an equation to solve this problem. Use n as the variable to represent the number.

20. Multiply the following binomials: (x + 5)(x - 5)

MATA02H3 Lecture 7: Prime/Composite Numbers & Eratosthenes Sieve

MATA02H3 Lecture Notes - Lecture 6: Prime Number, Euclidean Algorithm, Linear Combination

MATA02H3 Lecture 7: Prime/Composite Numbers & Eratosthenes Sieve

MATA02H3 Lecture 8: Composite Consecutive Natural Numbers, Twin Primes, Positive Divisors and Euler (Φ) Function

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