Euler Enrichment Stage
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1 Euler Enrichment Stage Table of Contents Chapter 1 Problems I Like Michael Evans 1 Chapter 2 Primes and Composites 3 Chapter 3 Least Common Multiple 8 Chapter 4 Highest Common Factor and the Euclidean Algorithm 11 Chapter 5 Arithmetic Sequences 16 Chapter 6 Figurate Numbers 21 Chapter 7 Congruences 26 Chapter 8 Problems I Like Sharing - Alice Thomas 31 Chapter 9 Find That Angle 34 Chapter 10 Counting Techniques 51 Chapter 11 The Pigeon-Hole Principle 58 Chapter 12 Problems I Enjoy Sharing - Bruce Henry 62 Solutions 64
2 Chapter 1 Problems I Like Michael Evans 1 In magic squares, the sum of the numbers in each row, each column and each diagonal is constant Find A, B, C, D, E in the following magic square 15 A B C 25 D E 2 (a) What two whole numbers, neither ending in zero, when multiplied together equal exactly ? (b) Repeat for Find three consecutive numbers such that the sum of the first and third is 18 4 A palindromic number is a number which remains the same when the digits are reversed For example, is a palindromic number What is the next largest palindromic number? 5 What two-digit number is twice the product of its digits? 6 Find the values of the letters, each of which stands for a particular but different digit + + F O R T Y T E N T E N S I X T Y 7 By replacing the asterisks with a selection of the four operational symbols (+,,, ), complete this equation to make a true statement: =13 8 A prime number is an integer greater than 1 whose only divisors are itself and is a prime What is the next year that is a prime? 1
3 9 What is the largest three-digit prime each of whose digits is a prime? 10 A perfect number equals the sum of its factors, excluding the number itself Since 6 = , 6 is a perfect number Find another 11 Divide the face of the clock into 3 parts with 2 straight lines so that the sums of the numbers in the 3 parts are equal A total of 642 digits was used in numbering the pages of a book How many pages did the book contain? 13 In the cells shown, place a ten-digit number such that the digit in the first cell indicates the total number of 0s in the entire number, the digit in cell 1 indicates the number of 1s in the number and so on, to the last cell Now try Problem 1 in the Euler Student Problems Book 2
4 Chapter 2 Primes and Composites A code is devised by Secret Sam Sam has decided that he will denote A by the smallest number with 1 factor, B the smallest number with 2 factors and so on He sets out his code as shown in the table (a) (b) A 1 B 2 C 4 D 6 E 16 Z Complete the table for the code Secret Sam decides that he would like to punctuate his code properly and use some mathematical symbols For example he denotes = with the smallest number with 39 factors What is this number? Factors and Primes The factors of 6 are: 1, 2, 3 and 6 The factors of 28 are: 1, 2, 4, 7, 14 and 28 The factors of 7 are: 1 and 7 A natural number, a, isafactor of a natural number, b, ifthereexistsa natural number, k, such that: b = ak If a number greater than 1 has only factors 1 and itself it is said to be a prime Among the first 100 numbers, the following 25 are prime: 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 To consider whether a number, n, isprimeonemethodistoconsiderall the numbers less than n and test if they are factors of n For example, if n = 97, 9 < n<10 Therefore, we could test the numbers: 2, 3, 4, 5, 6, 7, 8 and 9 However, we do not need to test numbers which are not prime, so we only need to test 2, 3, 5 and 7 3
5 It is found that none of these numbers is a factor of 97 We have found 97 to be prime There are infinitely many primes Composites Anumber,m, is called a composite number if it can be written as a product: m = a b where a and b are numbers greater than 1 and less than m For example, 72 = 8 9= = = = = = This method of expressing a composite number in terms of its prime numbers is called prime decomposition For each number there is a unique prime decomposition One method of determining the prime decomposition is the following The process is repeated division Consider the number Therefore = The number of factors of a number can be determined through the prime decomposition of that number They can be systematically listed as shown: 4
6 This covers all combinations Thus there are = 64 factors of Alternatively, we note that the prime decomposition of is and that 2 7 has 8 factors, 3 3 has4factorsand7 1 has 2 factors Hence has factors This is investigated further in the Exercises of this chapter Exercises 1 Give the prime decomposition of Give the prime decomposition of (a) State the factors of 16 = 2 4 (note: include 1 and 16) (b) How many factors are there of 16? (c) State the factors of 256 = 2 8 (d) How many factors are there of 256? (e) State the number of factors of 2 n? 4 (a) State the factors of 125 = 5 3 (b) How many factors are there of 5 n? 5 (a) Give the prime decomposition of (b) Howmanyfactorsdoesthisnumberhave? 6 A number has a prime decomposition of How many factors does this number have? 7 Two and three differ by one and are both prime numbers What is the next such pair? 8 Find a prime number which is one less than a perfect square Is there another such number? 5
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