MACM 101 Lecture 25: Lecture 25 Part 3_ Integers

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browncricket987

21 Dec 2018

School

Simon Fraser University

Department

Math & Comp Sci

Course

MACM 101

Professor

Andrei Bulatov

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Document Summary

Every integer n (except for 1 and -1) has at least 2 positive divisors, 1 and n (or -n). A positive number that does not have any other positive divisor is called prime. Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, . Mersenne numbers are the numbers of the form mn = 2n-1. There are many prime numbers among mersenne numbers. The greatest known prime number is m77232917 = 277232917 - 1. The next candidate is mm61 = 22305843009213693951 - 1. A positive number that is not prime is called composite. Every composite number has a prime divisor. Let s be the set of all composite numbers that do not have a prime divisor. Since s n, by the well-ordering principle, it has a least element r. As r is not prime, it has a divisor, therefore, r = uv for some positive integers u and v.

Related Questions

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

MACM 101 Lecture 25: Lecture 25 Part 3_ Integers

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