LLG

Advanced Math and Science Pilot Class

Mathematics, Grade 10

Paris-Abu Dhabi

October 2009

Lesson : Arithmetic. In this chapter, all numbers are positive whole numbers. 1) Denition A multiple is a number made by multiplying together two other numbers. Example : Denition A factor or a divisor is a number which divides into another number without remainder. Example : Remark : Denition An even number is a whole number which, when divided by two, has no remainder. Any number which ends in 0,2,4,6,8 must be even. An odd number is a whole number which, when divided by two, has remainder of 1. Any number which ends in 1,3,5,7,9 must be odd. 2) Denition A prime number is a number having two, and only two dierent factors : 1 and itself. Remarks : • 1 is not a prime number since it only has one factor. • 2 is the only prime even number. • There is an innity of prime numbers. Let specify all the prime number less than 100, using the Eratosthenes'sieve, it is an algorithm for nding prime numbers. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 The numbers left are the prime numbers. 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 Proposition To know if a number is prime, I try to divide it by the prime numbers less than its square root. Example : specify if 737 is a prime number or not.

Fundamental theorem of arithmetic : writing a number as a product of primes =prime decomposition of whole number=prime factorization

Any number can be made by multiplying together a set of prime numbers and this can be done in only one way ; the reordering of the prime numbers is not considered as dierent. Example : 12=

Tree-diagram representing the set of factors of a number e.g. : 12

3) Denition Greatest Common Divisor The Greatest Common Divisor (GCD or HCF highest Common Factor) of two (or more) numbers is the common factor of all those numbers which has the greatest value. Example : the GCD of 10 and 12 is the GCD of 12 and 17 is Proposition The GCD of two numbers written as a product of primes is the product of the common primes with the lowest power. e.g. :10= 12= GCD(10 ;12)=

Application : to simplify a fraction in one step the more than possible (=to get the reduced form of a fraction), I divide the numerator and the denominator by their GCD. 10 Example : = 12 4) Denition Lowest Common Multiple The Lowest Common Multiple (LCM) of two (or more) numbers is the smallest possible number into which all of them will divide exactly. Example : the LCM of 10 and 12 is Proposition The LCM of two numbers written as a product of primes is the product of all the primes with the greatest power. e.g. :10= 12= LCM(10 ;12)=

Application : To add two fractions, they must have the same denominator, the easiest common denominator is the LCM. 1 1 Example : + = 10 12

2