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Chapter 5 Number Theory 5.1 Primes, composites, and tests for divisibility Primes and Composites Prime number (prime) – a counting number with exactly two different factors Composite number (composite) – a counting number with more than two factors Primes 2,3,5,7,11 only have themselves and 1 as factors Factor Trees Prime factorizaton – each tree takes you to 60+2*2*3*5 Fundamental Theorem of Arithmetic Each composite number can be expressed as the product of primes in exactly one way (except for the order of the factors). Example: express 180 as the product of primes 10*18 = 2*5*2*3*3 = 22*32*5 Divides Let a and b be any whole numbers with a ≠ 0 . We say that a divides b, and write a b if and only if there is a whole number x such that ax=b. The symbol a b means that a does not divide b Divides (cont) a divides b if and only if a is a factor of b. When a divides b, we can also say that a is a divisor of b, a is a factor of b, b is a multiple of a, and b is divisible by a. 3 12? 216 IS A MULTIPLE OF 6 ? 8 IS A DIVISOR OF 96 ? Test for divisibility by 2,5, and 10 A number is divisible by 2 if and only if its ones digit is 0,2,4,6, or 8. A number is divisible by 5 if and only if its ones digit is 0 or 5 A number is divisible by 10 if and only if its ones digit is 0 Theorem: Let a, m, n, and k be whole numbers where a ≠ 0 a) If a m and a n then a (m+n) b) If a m and a n then a (m-n) for c) If a m, then a km. 3 (9+12) m≥n Tests for divisibility by 4 and 8 A number is divisible by 4 if and only if the number represents by its last two digits is divisible by 4 A number is divisible by 8 if and only if the number represented by its last three digits is divisible by 8 4 1432 ? 8 98,765,432 ? Tests for divisibility by 3 and 9 A number is divisible by 3 if and only if the sum of its digits is divisible by 3 A number is divisible by 9 if and only if the sum of its digits is divisible by 9 3 12,345 ? 9 6543 ? Test for Divisibility by 11 A number is divisible by 11 if and only if 11 divides the difference of the sum of the digits whose place values are odd powers of 10 and the sum of the digits whose place values are even powers of 10 11 5346 11 909,381 Test for Divisibility by 6 A number is divisible by 6 if and only if both of the tests for divisibility by 2 and 3 hold. A number is divisible by 2 if and only if its ones digit is 0,2,4,6, or 8. A number is divisible by 3 if and only if the sum of its digits is divisible by 3 Theorem: A number is divisible by the product, ab, of two nonzero whole numbers a and b if it is divisible by both a and b, and a and b have only the number 1 as a common factor. Prime Factor Test To test for prime factors of a number n, one need only search for prime 2 factors p of n, where p ≤ n( or _ p ≤ n ) Determine whether the following are prime 299 401 Section 5.2 Counting Factors Greatest Common Factor Least Common Multiple Theorem Suppose that a counting number n is expressed as a product of distinct primes with their respective exponents, say n = ( p1n )( p2n )...( pmn ) Then the number of factors of n is the product (n1 + 1) * (n2 + 1)... (nm + 1) 1 2 m (The number of factors depends on the exponents NOT Prime factors) . Greatest Common Factor The greatest common factor (GCF) of two (or more) nonzero whole numbers is the largest whole number that is a factor of both (all) of the numbers. The GCF of a and b is written GCF(a,b). . GCF – Set intersection method 1. Find all factors GCF of 24 and 36 24=23*3 then 4*2=8 factors and 36 = 22*32 then 3*3=9 factors of 36 Factors for 24 = ? Factors for 36 = ? 2. {1,2,3,4,6,8,12,24} {1,2,3,4,6,9,12,18,36} Find all common factors by taking the intersection of the two sets from step 1 {1,2,3,4,6,8,12,24} I {1,2,3,4,6,9,12,18,36} = {1,2,3,4,6,12} 3. Find the largest number in the set of common factors in step 2 Therefore 12 is the GCF of 24 and 36 . GCF- Prime Factorization Method 1. Express the numbers in their prime factor exponential form 24=23*3 and 36=22*32 2. The GCF will be the number 2m3n where m is the smaller of the exponents of the 2s and n is the smaller of the exponents of the 3s For 23*3 and 22*32 m is the smaller of 3 and 2 and n is the smaller of 1 and 2 Then GCF is 22*31=12 . Theorem If a and b are whole numbers with a ≥ b, then GCF(a,b) = GCF(a – b,b) Find the GCF(546,390) GCF(546,390) = GCF(546-390,390) = GCF(156,390) = GCF(390-156,156) = GCF(234,156) =GCF(234-156,156) = GCF(78,156) = GCF (156-78,78) = GCF (78,78) . Euclidean Algorithm If a and b are whole numbers, with a ≥ b and a ≥ bq + r where r < b then GCF(a,b)=GCF(r,b) To find the GCF of any two numbers this theorem can be applied repeatedly until a remainder of zero is obtained. The final divisor that leads to the zero remainder is the GCF . Euclidean Algorithm 4 R 72 840 3432 11R 48 72 840 1R 24 48 72 2 R0 24 48 Least Common Multiple The least common multiple (LCM) of two (or more) nonzero whole numbers is the smallest nonzero whole number that is a multiple of each (all) of the numbers. Written as LCM(a,b) Useful when adding or subtracting fractions LCM – Set Intersection Method 1. List the first several nonzero multiples of 24 and 36 Multiples of 24 {24,48,72,96,120,144…} Multiples of 36 {36,72,108,144,…} 2. Find the first several common multiples of 24 and 36 by taking the intersection of the sets {24,48,72,96,120,144,...} I {36,72,108,144,... = {72,144,...} 3. Find the smallest number in the union from step 2 = LCM (72) LCM – Prime Factorization method 1. Express the numbers in their prime factor exponential form 24=23*3 and 36=22*32 2. The LCM will be the number 2r3s where r is the larger of the exponents of the twos and s is the larger of the exponents of the threes Then r is 3 and s is 2 then the LCM is 23*32=72 LCM – Build-up Method 1. Express the numbers in their prime factor exponential form 24=23*3 and 36=22*32 2. Select the prime factorization of one of the numbers and build the LCM by taking the highest exponent of each factor Begin with 24=23*3 and compare the 2’s factor with 36=22*32 and take the largest 23 Then take the prime factor 3 and compare with the 3’s factor of 32 and take the largest – 32 Build up to 23*32* = LCM=72