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Thinking Critically 4.1 Divisibility Of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples Copyright © 2012, 2009, and 2006, Pearson Education, Inc. 4.1 Divisibility and Natural Numbers Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-2 DEFINITION: DIVIDES, FACTORS, DIVISOR, MULTIPLE If a and b are whole numbers with b ≠ 0 and there is a whole number q such that a = bq, we say that b divides a. We also say that b is a factor of a or a divisor of a and that a is a multiple of b. If b divides a and b is less than a, it is called a proper divisor of a. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-4 DEFINITION: EVEN AND ODD WHOLE NUMBERS A whole number a is even precisely when it is divisible by 2. a = 2k for a whole number k. A whole number b that is not even is called an odd number. b = 2j + 1 for a whole number j. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-5 DEFINITION: PRIMES, COMPOSITE NUMBERS, UNITS A natural number that possesses exactly two different factors, itself and 1, is called a prime number. A natural number that possesses more than two different factors is called a composite number. The number 1 is called a unit; it is neither prime nor composite. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-6 THE FUNDAMENTAL THEOREM OF ARITHMETIC (Simple-Product Form) Every natural number greater than 1 is a prime or can be expressed as a product of primes in one, and only one, way apart from the order of the prime factors. NOTE: This is why we do not think of 1 as either a prime or composite number, to preserve uniqueness of the product. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-7 Example 4.3 Prime Factors of 600 Write 600 as a product of primes. USING A FACTOR TREE USING SHORT DIVISION 600 = 2 • 2 • 2 • 3 • 5 • 5 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-8 THE FUNDAMENTAL THEOREM OF ARITHMETIC (Prime-Power Form) Every natural number n greater than 1 is a power of a prime or can be expressed as a product of powers of different primes in one, and only one, way apart from order. This representation is called the primepower representation of n. 600 = 2 • 2 • 2 • 3 • 5 • 5 = 23 • 31 • 52 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-9 PRIMES The Number of Primes There are infinitely many primes. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-12 4.2 Tests for Divisibility Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-10 DIVISIBILITY OF SUMS AND DIFFERENCES Let d, a, and b be natural numbers. Then if d divides both a and b, then it also divides their sum, a + b, and their difference, a – b. Example: 3 divides both 36 and 15, thus it also divides 36 + 15 = 51 and 36 – 15 = 21. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-14 TESTS FOR DIVISIBILITY By 2: A natural number is divisible by 2 exactly when its base ten units digit is 0, 2, 4, 6, or 8. By 5: A natural number is divisible by 5 exactly when its base ten units digit is 0 or 5. By 10: A natural number is divisible by 10 exactly when its base ten units digit is 0. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-15 TESTS FOR DIVISIBILITY By 4: A natural number is divisible by 4 when the number represented by its last two digits is divisible by 4. By 8: A natural number is divisible by 8 when the number represented by its last three digits is divisible by 8. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-16 TESTS FOR DIVISIBILITY Is 81,164 divisible by 4 and by 8? 81,164 is divisible by 4 since 64 is divisible by 4. 81,164 is not divisible by 8 since 164 is not divisible by 8. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-17 TESTS FOR DIVISIBILITY By 3: A natural number is divisible by 3 if and only if the sum of its digits is divisible by 3. By 9: A natural number is divisible by 9 if and only if the sum of its digits is divisible by 9. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-18 TESTS FOR DIVISIBILITY Is 81,165 divisible by 3 and by 9? Sum of digits: 8 + 1 +1 + 6 + 5 = 21 81,164 is divisible by 3 since 21 is divisible by 3. 81,164 is not divisible by 9 since 21 is not divisible by 9. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-19 TESTS FOR DIVISIBILITY By 11: A natural number is divisible by 11 exactly when the sum of its digits in the even and odd positions have a difference that is divisible by 11. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-20 Example 4.9 Test for divisibility by 11 Is 42,315,690 divisible by 11? Even positions: 2 + 1 + 6 + 0 = 9 Odd positions: 4 + 3 + 5 + 9 = 21 Difference: 21 – 9 = 12 which is not divisible by 11. It follows that 42,315,690 is not divisible by 11. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-21 4.3 Greatest Common Divisors and Least Common Multiples Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-19 DEFINITION: GREATEST COMMON DIVISOR Let a and b be whole numbers not both 0. The greatest natural number d that divides both a and b is called their greatest common divisor and we write d GCD(a, b ). Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-26 Example 4.12 Finding the GCD by Intersection of Sets Find the greatest common divisor of 24 and 27. (Rainbow) List the sets of divisors of each number. D24 {1, 2,3, 4, 6,8,12, 24} D27 {1,3,9, 27} Find the intersection of these sets. D24 D27 {1,3} GCD(24, 27) 3 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-27 FINDING THE GCD: PRIME FACTORIZATION METHOD Let a and b be natural numbers. Then the GCD(a, b) is the product of the prime powers in the prime-power factorizations of a and b which have the smaller exponents (including zero). Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-28 FINDING THE GCD: PRIME FACTORIZATION METHOD Compute the greatest common divisor of m 2 3 5 and n 3 5 7. 2 3 4 Rewrite the numbers as follows: m 2 3 5 7 and n 2 3 5 7 1 2 3 0 0 4 1 1 GCD(m, n) 20 32 51 70 9 5 45 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-29 DEFINITION: LEAST COMMON MULTIPLE Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their least common multiple, and we write m LCM(a, b). Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-32 Example 4.15 Finding a LCM by Set Intersections Find the least common multiple of 9 and 15. List the sets of multiples of each number. M 9 {9,18, 27,36, 45,54,63,72,81,90,...} M15 {15,30, 45,60, } Find the intersection of these sets. M 9 M15 {45,90, } LCM(9,15) 45 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-33 FINDING THE LCM: PRIME FACTORIZATION METHOD Let m and n be natural numbers. Then the LCM(m,n) is the product of the prime powers in the prime-power factorizations of m and n that have the larger exponents. Copyright © 2012, 2009, and 2006, Pearson Education, Inc. Slide 4-34 Example Finding the LCM: Prime-Power Method Compute the least common multiple of 2 3 4 m 2250 2 3 5 and n 2835 3 5 7. Rewrite the numbers as follows: m 2 3 5 7 and n 2 3 5 7 1 2 3 0 0 4 1 1 LCM(m, n) 2 3 5 7 141,750 1 4 Copyright © 2012, 2009, and 2006, Pearson Education, Inc. 3 1 Slide 4-35