DOCSLIB.ORG

Numbers and Then a Space Before the Next Set of Digits

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Numbers and Then a Space Before the Next Set of Digits 1 NUMBER SENSE AND NUMERATION REPRESENTING NUMBER 1. WRITTEN FORM Examples: i. One ii. Twelve iii. Six hundred eight iv. Seven thousand eight three v. Seven hundredths vi. Four hundred thousandths vii. Thirty two thousand, four hundred ninety six viii. Nine hundred five billion, six hundred fifteen million, two hundred eight thousand, five hundred eight. 2 2. STANDARD FORM Information: Beginning with the smallest whole number (the ones), there are THREE digits in each grouping of numbers and then a space before the next set of digits. For the largest place value, there may be one, two or three digits. Zero is a digit which represents a place value unless it is at the beginning of a whole number, and then it just a place holder Examples: i. 32 469 ii. 905 615 208 508 3 3. EXPANDED FORM Information: Numbers written in standard form can be expanded out to show the value of each digit When you do this, you what each digit in the number actually represents numerically (its place value) 2 6 7 5 9 8 3 TENS ONES MILLIONS HUNDREDS THOUSANDS OF THOUSANDS TENS OF THOUSANDS TENS OF HUDREDS Examples: i. 8 765 = 8 000 + 700 + 60 + 5 ii. 943 567 832.23 = 900 000 000 + 40 000 000 + 3 000 000 + 5 00 000 + 60 000 + 7 000 + 800 + 30 + 2 + 0.2 + 0.03 4 PLACE VALUE Information: our number system has ten digits. They are: 0, 1, 2 , 3, 4, 5, 6, 7, 8, and 9 How those digits are arranged makes a number. It is the POSITION of the digit in the number that gives the digit a value. The further to the right a digit is in a number) the larger the value of the digit is. The further to the left a digit is (including in the decimal portion) the smaller the value of the digit is. Example: 88 888.88 Even though all of the digits are 8, each place value means the 8s are worth different values. The underlined 8 is in the ten thousand place value position. It represents 8 groups of 10 000 or 80 000. The boxed digit 8 is in the hundredths place value position. It represents 8 groups of a hundredth, or .08 5 EXPONENTS 2 EXPONENT POWER 5 The exponent shows how many times you must multiple the base number times itself. Examples: 52 means 5 x5, which equals 25 53 means 5 x 5 x 5, which equals 125 54 means 5 x 5 x 5 x 5 = 625 6 Notice how the exponent POWERS OF TEN and the number of zeros There is NO power of one! in the number are the same! MATH IS ALL 102 = 10 x 10 = 100 ABOUT PATTERNS 103 = 10 x 10 x 10 = 1 000 104 = 10 x 10 x 10 x 10 = 10 000 105 = 10 x 10 x 10 x 10 x 10 = 100 000 106 = 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100 000 000 109 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 000 10 10 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 000 7 EXPANDED FORM USING POWERS OF TEN Since we have looked at Powers of ten, the next step is to show numbers in expanded form using powers of ten! Example: 45 678 The digit 4 represents 4 groups of 10 000, or 40 000. We know that 104 is 10 000 We can show 40 000 as 4 x 104 The digit 5 represents 5 groups of 1 000 or 5 000 We know that 103 is 1 000 We can show 5 000 as 5 x 103 So, in expanded for, using Power of Ten: 45 678 = 4 x 104 + 5 x 103 + 6 x 102 + 70 + 8. 8 DIVISIBILITY RULES Divisible by…? The Strategy Example 2 Must end in an even number 252, 756, 928, 4 354 (o, 2 , 4, 6, 8) 3 Add all digits together until they form 76 825 either a single digit or a digit you 7+6+8+2+5 = 28 recognize as being divisible by 3. If the sum of the digits is divisible by three, 2+8 = 10 then so is the number 10 is NOT divisible by 3, so neither is 76 825. 4 Look at the last 2 digits. If they are 9 6 5 7 57 is not divisible by 2, then so is the number. divisible by 4, so neither is 9 657. 5 Any number that ends in a zero or a five 6 890 is divisible by 5 is divisible by 5 because is ends in a zero. 6 Both rules for 2 and 3 apply 3 246 – last digit is even, so divisible by 2 and sum of digits is 15 which is divisible by 3, so it is divisible by 6 9 Sum of digits is divisible by 9 48 456 4+8+4+5+6 = 27 27 is divisible by 9, so so is 48 456 10 Any number that ends in a zero is 56 090 – is divisible by divisible by 10 10 9 MULTIPLES Think of your multiplication tables… Multiples of a number are the product of that number times any other number. So, for example, the first 4 multiples of 2 are: 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 The first 6 multiples of 3 are: 3, 6, 9, 12, 15, 18, but 300 and 252 are also multiples of 3 because they can be divided by 3 with no remainder. 10 FACTORS Factors are one of two or more numbers that, when multiplied together, produce a give product. So, for example, the factors of 6 are 2 and 3 because 2 x 3 = 6. Sometimes we are asked to find ALL of the factors of a number. That means all of the possible combinations of numbers that we could multiply together to give a specific product. EXAMPLE: Because ALL numbers can be made by multiplying the number by 1, we do NOT 36 include 1 as a factor of a number All of the factors of 36 are: (1 x 36) 2 x 18 3 x 12 4 x 9 6 x 6 The factors of the number 36 are: 2, 3, 4, 6, 9, 12, 18 and 36. 11 FACT FAMILIES – INVERSE OPERATIONS You will be very surprised how often you find that fact families are helpful as you progress through intermediate and high school math! You learned about fact families in primary grades, but guess what? You apply (use) this knowledge in all different forms of more challenging math! Fact families show opposite or INVERSE operations. Here are two examples of a fact families. 5 + 8 = 13 4 + 3 = 12 8 + 5 = 13 3 + 4 = 12 13 – 8 = 5 12 ÷ 3 = 4 13 – 5 = 8 12 ÷ 4 = 3 Fact families show how numbers are related. They teach us that we can use the inverse operation to help us. The inverse operation of addition is subtraction and the inverse operation of multiplication is division. 12 OPERATIONS IN MATH There are 4 basic operations in math. The answers you get by using different operations have different names! The answer to addition is called the SUM The answer to subtraction is called the DIFFERENCE The answer to multiplication is called the PRODUCT The answer to division is called the QUOTIENT 13 “FRIENDLY” NUMBERS Okay, well, numbers are not really either friendly or unfriendly, but some numbers are a bit easier to work with than others, so we call them, “friendly numbers.” Examples of friendly numbers include any multiple of 10, because with a bit of practise, these numbers (because they end in zeroes) are easy to add, subtract, multiply and divide. What is one person’s idea of a friendly number may differ from another’s, but usually, the following are considered, “friendly:” 2, 5, 10, 20, 25, 50, 100, 1 000 14 INTEGERS – POSITIVE AND NEGATIVE NUMBERS An integer is a WHOLE number that is not a fraction. Our number system is INFINITE which means that it never ends. You can ALWAYS add to or take away from a number. Numbers that are BELOW (LESS THAN) zero are called NEGATIVE integers (or numbers!). Numbers that are ABOVE (GREATER THAN) zero are called POSITIVE integers (or numbers!). -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 We encounter negative numbers all the time in our real lives. In the winter, in Ontario, the temperature is often below zero! We would show 25° below zero as -25°celcius. If you are below sea level, then this is measured as below zero. This is below sea level! 15 PRIME AND COMPOSITE NUMBERS We describe whole numbers in different ways. One of the ways is by looking at how many factors a whole number has. PRIME NUMBERS Some numbers only have 2 factors, one and the number itself. These numbers are called PRIME NUMBERS. Think about a glass plate made of a number. If you throw it up and it can only break into 2 sections, (1 and the number itself), it is PRIME.
Load more

Recommended publications
  • Connected Components of Complex Divisor Functions
    CONNECTED COMPONENTS OF COMPLEX DIVISOR FUNCTIONS COLIN DEFANT Princeton University Fine Hall, 304 Washington Rd. Princeton, NJ 08544 X c Abstract. For any complex number c, define the divisor function σc : N ! C by σc(n) = d . djn Let σc(N) denote the topological closure of the range of σc. Extending previous work of the current author and Sanna, we prove that σc(N) has nonempty interior and has finitely many connected components if <(c) ≤ 0 and c 6= 0. We end with some open problems. 2010 Mathematics Subject Classification: Primary 11B05; Secondary 11N64. Keywords: Divisor function; connected component; spiral; nonempty interior; perfect number; friendly number. 1. Introduction One of the most famous and perpetually mysterious mathematical contributions of the ancient Greeks is the notion of a perfect number, a number that is equal to the sum of its proper divisors. Many early theorems in number theory spawned from attempts to understand perfect numbers. Although few modern mathematicians continue to attribute the same theological or mystical sig- nificance to perfect numbers that ancient people once did, these numbers remain a substantial inspiration for research in elementary number theory [2, 3, 5, 11, 14{16, 18]. arXiv:1711.04244v3 [math.NT] 4 May 2018 P A positive integer n is perfect if and only if σ1(n)=n = 2, where σ1(n) = djn d. More generally, for any complex number c, we define the divisor function σc by X c σc(n) = d : djn These functions have been studied frequently in the special cases in which c 2 {−1; 0; 1g.
    [Show full text]
  • MAKING SENSE of MATHEMATICS THROUGH NUMBER TALKS: a CASE STUDY of THREE TEACHERS in the ELEMENTARY CLASSROOM Under the Direction of William O
    MAKING SENSE OF MATHEMATICS THROUGH NUMBER TALKS: A CASE STUDY OF THREE TEACHERS IN THE ELEMENTARY CLASSROOM by MIRANDA GAYLE WESTBROOK A Dissertation Submitted to the Faculty in the Curriculum and Instruction Program of Tift College of Education at Mercer University in Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY Atlanta, GA 2019 MAKING SENSE OF MATHEMATICS THROUGH NUMBER TALKS: A CASE STUDY OF THREE TEACHERS IN THE ELEMENTARY CLASSROOM by MIRANDA GAYLE WESTBROOK Approved: ________________________________________________________________________ William O. Lacefield, III, Ed.D. Date Dissertation Committee Chair ________________________________________________________________________ Justus J. Randolph, Ph.D. Date Dissertation Committee Member ________________________________________________________________________ Jeffrey S. Hall, Ed.D. Date Dissertation Committee Member ________________________________________________________________________ Jane West, Ed.D. Date Director of Doctoral Studies, Tift College of Education ________________________________________________________________________ Thomas R. Koballa, Jr., Ph.D. Date Dean, Tift College of Education DEDICATION To my loving husband, Chad, I am forever indebted to the unconditional love, support, and patience you bestowed as I engaged in this endeavor. You always encouraged me to persevere, and together, we transformed this dream into a reality. I am truly blessed to have you in my life. To my mom and dad, you inspired me to follow my dreams and taught me the value of hard work. The morning phone calls were always uplifting, and I love you both with all my heart. To Brenda, thank you for understanding and supporting me on this journey. To Linda, you were always confident that I would achieve this goal. Your enthusiasm and interest in my work gave me the strength I needed to keep writing.
    [Show full text]
  • 2012 Volume 20 No.2-3 GENERAL MATHEMATICS Daniel Florin SOFONEA Ana Maria ACU Dumitru ACU Heinrich Begehr Andrei Duma Heiner
    2012 Volume 20 No.2-3 GENERAL MATHEMATICS EDITOR-IN-CHIEF Daniel Florin SOFONEA ASSOCIATE EDITOR Ana Maria ACU HONORARY EDITOR Dumitru ACU EDITORIAL BOARD Heinrich Begehr Andrei Duma Heiner Gonska Shigeyoshi Owa Vijay Gupta Dumitru Ga¸spar Piergiulio Corsini Dorin Andrica Malvina Baica Detlef H. Mache Claudiu Kifor Vasile Berinde Aldo Peretti Adrian Petru¸sel SCIENTIFIC SECRETARY Emil C. POPA Nicu¸sorMINCULETE Ioan T¸INCU EDITORIAL OFFICE DEPARTMENT OF MATHEMATICS AND INFORMATICS GENERAL MATHEMATICS Str.Dr. Ion Ratiu, no. 5-7 550012 - Sibiu, ROMANIA Electronical version: http://depmath.ulbsibiu.ro/genmath/ Contents I. A. G. Nemron, A curious synopsis on the Goldbach conjecture, the friendly numbers, the perfect numbers, the Mersenne composite numbers and the Sophie Germain primes . .5 A. Ayyad, An investigation of Kaprekar operation on six-digit numbers computational approach . 23 R.Ezhilarasi,T.V. Sudharsan,K.G. Subramanian, S.B.Joshi,A subclass of harmonic univalent functions with positive coefficients defined by Dziok-Srivastava operator . 31 G. Akinbo, O.O. Owojori, A.O. Bosede, Stability of a common fixed point iterative procedure involving four selfmaps of a metric space . 47 S. Rahrovi, A. Ebadian, S. Shams , G-Loewner chains and parabolic starlike mappings in several complex variables . 59 G. I. Oros , A class of univalent functions obtained by a general multi- plier transformation . 75 Z. Tianshu, Legendre-Zhang's Conjecture & Gilbreath's Conjecture and Proofs Thereof . 87 M. K. Aouf, A certain subclass of p-valently analytic functions with negative coefficients . 103 H. Jolany, M. Aliabadi, R. B. Corcino, M.R.Darafsheh, A note on multi Poly-Euler numbers and Bernoulli polynomials .
    [Show full text]
  • The Love of Numbers
    Journal and Proceedings of The Royal Society of New South Wales Volume 119 pts 1-2 [Issued December, 1986] pp.95-101 Return to CONTENTS The Love of Numbers* John H. Loxton [Presidential Address, April 1986] It is now well known that the answer to the ultimate question of life, the universe and everything is 42. [1.] So we see that numbers are the fundamental elements of civilisation as we know it. Numbers such as telephone numbers and car licenses serve to whip our activities into some sort of order. Numbers are turned to good account by the Gas Board and the Taxation Office. Numbers, especially big round ones, fuel the arguments of economists and politicians. Numbers have mystical properties: 7 is a nice friendly number, while 13 is an unlucky one, particularly on Fridays. Although we would not rationally except to get anything significant by adding Margaret Thatcher’s telephone number to Bob Hawke’s this is still a very popular method of prophecy. For example, in the prophecy of Isaiah, the lion proclaims the fall of Babylon because the numerical equivalents of the Hebrew words for “lion” and “Babylon” have the same sum. [2.] Numbers are ubiquitous. All this was much more pithily expressed by Leopold Kronecker in 1880: “God created the integers – all else is the work of man”. Mathematics is the numbers game par excellence. This is not to say that mathematicians are better than anybody else at reconciling their bank statements. In fact, Isaac Newton, who was Master of the Mint, employed a book-keeper to do his sums.
    [Show full text]
  • The Impact of Regular Number Talks on Mental Math Computation Abilities
    St. Catherine University SOPHIA Masters of Arts in Education Action Research Papers Education 5-2014 The Impact of Regular Number Talks on Mental Math Computation Abilities Anthea Johnson St. Catherine University, [email protected] Amanda Partlo St. Catherine University, [email protected] Follow this and additional works at: https://sophia.stkate.edu/maed Part of the Education Commons Recommended Citation Johnson, Anthea and Partlo, Amanda. (2014). The Impact of Regular Number Talks on Mental Math Computation Abilities. Retrieved from Sophia, the St. Catherine University repository website: https://sophia.stkate.edu/maed/93 This Action Research Project is brought to you for free and open access by the Education at SOPHIA. It has been accepted for inclusion in Masters of Arts in Education Action Research Papers by an authorized administrator of SOPHIA. For more information, please contact [email protected]. The Impact of Regular Number Talks on Mental Math Computation Abilities An Action Research Report By Anthea Johnson and Amanda Partlo The Impact of Regular Number Talks on Mental Math Computation Abilities By Anthea Johnson and Amanda Partlo Submitted May of 2014 in fulfillment of final requirements for the MAED degree St. Catherine University St. Paul, Minnesota Advisor _______________________________ Date _____________________ Abstract The purpose of our research was to determine what impact participating in regular number talks, informal conversations focusing on mental mathematics strategies, had on elementary students’ mental mathematics abilities. The research was conducted in two urban fourth grade classrooms over a two month period. The data sources included a pre- and post-questionnaire, surveys about students’ attitudes towards mental math, a pretest and a posttest containing addition and subtraction problems to be solved mentally, teacher reflective journals, and student interviews.
    [Show full text]
  • Integer Sequences
    UHX6PF65ITVK Book > Integer sequences Integer sequences Filesize: 5.04 MB Reviews A very wonderful book with lucid and perfect answers. It is probably the most incredible book i have study. Its been designed in an exceptionally simple way and is particularly just after i finished reading through this publication by which in fact transformed me, alter the way in my opinion. (Macey Schneider) DISCLAIMER | DMCA 4VUBA9SJ1UP6 PDF > Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular
    [Show full text]
  • Relationship Between Prime Factors of a Number and Its Categorization As Either Friendly Or Solitary
    RELATIONSHIP BETWEEN PRIME FACTORS OF A NUMBER AND ITS CATEGORIZATION AS EITHER FRIENDLY OR SOLITARY RICHARD KARIUKI GACHIMU MASTER OF SCIENCE (Pure Mathematics) JOMO KENYATTA UNIVERSITY OF AGRICULTURE AND TECHNOLOGY 2012 Relationship Between Prime Factors of a Number and its Categorization as Either Friendly or Solitary Richard Kariuki Gachimu A thesis submitted in partial fulfilment for the Degree of Master of Science in Pure Mathematics in the Jomo Kenyatta University of Agriculture and Technology 2012 DECLARATION This thesis is my original work and has not been presented for a degree in any other University. Signature…………..……………..………..… Date…………………….…………. eeeeeeeeeeeeeRichard Kariuki Gachimu This thesis has been submitted for examination with our approval as University Supervisors. Prof. Cecilia W. Mwathi (Deceased) JKUAT, Kenya Signature…………..………..……………… Date………...……………………… ttttttttttttttttttttt Dr. Ireri N. Kamuti tttttttttttttttt Kenyatta University, Kenya ii DEDICATION Little did I know of the significance of the day, not only to my life but also to the lives of many others. A new dawn had come; a different kind of light had shone, light that could turn around lives. The truth, however bitter, had finally come; truth that was intended to set the lives of many free. Thursday 12th June, 2003 marked the beginning of a new beginning and unto this day I dedicate this work. iii ACKNOWLEDGEMENT I am most thankful to my Living God, creator of heavens and earth, for His immeasurable love, mercy, kindness, tender care, favour and all sorts of blessings He has showered upon me. He has seen me through this tough and demanding journey. It is His will that I am what I am and my destiny rests upon Him.
    [Show full text]
  • 25 Primes in Arithmetic Progression
    b2530 International Strategic Relations and China’s National Security: World at the Crossroads This page intentionally left blank b2530_FM.indd 6 01-Sep-16 11:03:06 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 SA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 K office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Ribenboim, Paulo. Title: Prime numbers, friends who give problems : a trialogue with Papa Paulo / by Paulo Ribenboim (Queen’s niversity, Canada). Description: New Jersey : World Scientific, 2016. | Includes indexes. Identifiers: LCCN 2016020705| ISBN 9789814725804 (hardcover : alk. paper) | ISBN 9789814725811 (softcover : alk. paper) Subjects: LCSH: Numbers, Prime. Classification: LCC QA246 .R474 2016 | DDC 512.7/23--dc23 LC record available at https://lccn.loc.gov/2016020705 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, SA. In this case permission to photocopy is not required from the publisher. Typeset by Stallion Press Email: [email protected] Printed in Singapore YingOi - Prime Numbers, Friends Who Give Problems.indd 1 22-08-16 9:11:29 AM October 4, 2016 8:36 Prime Numbers, Friends Who Give Problems 9in x 6in b2394-fm page v Qu’on ne me dise pas que je n’ai rien dit de nouveau; la disposition des mati`eres est nouvelle.
    [Show full text]
  • The Form of the Friendly Number of 10
    The Form Of The Friendly Number Of 10 Sourav Mandal Abstract 9 Any positive integer 푛 other than 10 with abundancy index ⁄5 must be a square with at least 6 distinct prime factors, the smallest being 5, and my new argument about the form of the friendly 25(52푐+1−1)(8푛+1−2푐) number of 10 is ,if exist. Futher,at least one of the prime factors must be 4 congruent to 1 modulo 3 and appear with an exponent congruent to 2 modulo 6 in the prime factorization of 푛. Introduction 휎(푛) For a positive integer 푛, the sum of the positive divisors of 푛 is denoted by 휎(푛); the ratio is 푛 known as the abundancy index of 푛 or sometimes the ratio denoted as 휎̂(푛) or 퐼(푛). A pair (푎, 푏) is called a friendly pair if휎̂(푎) = 휎̂(푏) in this case,it is also common to say that 푏 is 푎 friend of a or simply that 푏 and 푎 are friend. Perfect numbers have abundancy index 2,and thus are all friendly numbers with abundancy index less than 2 are often called deficient, while numbers whose abundancy index are greater than 2 are called abundant. The original problem was to show that the density of friendly integers is ℕ is unity and the density of solitary numbers (numbers with no friends) is zero. 휎(푛) We used two main approaches: one was an analysis of the function. While the other used 푛 number theoretic arguments to find a representation for a friend of 10.
    [Show full text]
  • Download PDF # Integer Sequences
    4DHTUDDXIK44 ^ eBook ^ Integer sequences Integer sequences Filesize: 8.43 MB Reviews Extensive information for ebook lovers. It typically is not going to expense too much. I discovered this book from my i and dad recommended this pdf to learn. (Prof. Gerardo Grimes III) DISCLAIMER | DMCA DS4ABSV1JQ8G > Kindle » Integer sequences INTEGER SEQUENCES Reference Series Books LLC Dez 2011, 2011. Taschenbuch. Book Condition: Neu. 247x192x7 mm. This item is printed on demand - Print on Demand Neuware - Source: Wikipedia. Pages: 141. Chapters: Prime number, Factorial, Binomial coeicient, Perfect number, Carmichael number, Integer sequence, Mersenne prime, Bernoulli number, Euler numbers, Fermat number, Square-free integer, Amicable number, Stirling number, Partition, Lah number, Super-Poulet number, Arithmetic progression, Derangement, Composite number, On-Line Encyclopedia of Integer Sequences, Catalan number, Pell number, Power of two, Sylvester's sequence, Regular number, Polite number, Ménage problem, Greedy algorithm for Egyptian fractions, Practical number, Bell number, Dedekind number, Hofstadter sequence, Beatty sequence, Hyperperfect number, Elliptic divisibility sequence, Powerful number, Znám's problem, Eulerian number, Singly and doubly even, Highly composite number, Strict weak ordering, Calkin Wilf tree, Lucas sequence, Padovan sequence, Triangular number, Squared triangular number, Figurate number, Cube, Square triangular number, Multiplicative partition, Perrin number, Smooth number, Ulam number, Primorial, Lambek Moser theorem,
    [Show full text]
  • A42 Integers 16 (2016) the Sum of Binomial Coefficients and Integer Factorization
    #A42 INTEGERS 16 (2016) THE SUM OF BINOMIAL COEFFICIENTS AND INTEGER FACTORIZATION Yingpu Deng Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P. R. China [email protected] Yanbin Pan Key Laboratory of Mathematics Mechanization, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, P.R. China [email protected] Received: 10/22/14, Revised: 1/18/16, Accepted: 6/3/16, Published: 6/10/16 Abstract The combinatorial sum of binomial coefficients n n n k (a) := a − i k r k i (mod r) ✓ ◆ ⌘ X has been studied widely in combinatorial number theory, especially when a = 1 and a = 1. In this paper, we connect it with integer factorization for the first time. − More precisely, given a composite n, we prove that for any a coprime to n there exists a modulus r such that the combinatorial sum has a nontrivial greatest common divisor with n. Denote by FAC(n, a) the least r. We present some elementary upper bounds for it and believe that some bounds can be improved further since FAC(n, a) is usually much smaller in the experiments. We also proposed an algorithm based on the combinatorial sum to factor integers. Unfortunately, it does not work as well as the existing modern factorization methods. However, our method yields some interesting phenomena and some new ideas to factor integers, which makes it worthwhile to study further. 1. Introduction Let n, i, r and a be integers with n > 0 and r > 0.
    [Show full text]
  • Tutorme Subjects Covered.Xlsx
    Subject Group Subject Topic Computer Science Android Programming Computer Science Arduino Programming Computer Science Artificial Intelligence Computer Science Assembly Language Computer Science Computer Certification and Training Computer Science Computer Graphics Computer Science Computer Networking Computer Science Computer Science Address Spaces Computer Science Computer Science Ajax Computer Science Computer Science Algorithms Computer Science Computer Science Algorithms for Searching the Web Computer Science Computer Science Allocators Computer Science Computer Science AP Computer Science A Computer Science Computer Science Application Development Computer Science Computer Science Applied Computer Science Computer Science Computer Science Array Algorithms Computer Science Computer Science ArrayLists Computer Science Computer Science Arrays Computer Science Computer Science Artificial Neural Networks Computer Science Computer Science Assembly Code Computer Science Computer Science Balanced Trees Computer Science Computer Science Binary Search Trees Computer Science Computer Science Breakout Computer Science Computer Science BufferedReader Computer Science Computer Science Caches Computer Science Computer Science C Generics Computer Science Computer Science Character Methods Computer Science Computer Science Code Optimization Computer Science Computer Science Computer Architecture Computer Science Computer Science Computer Engineering Computer Science Computer Science Computer Systems Computer Science Computer Science Congestion Control
    [Show full text]