Thinking Critically 4.1 Divisibility Of Natural Numbers 4.2 Tests for Divisibility 4.3 Greatest Common Divisors and Least Common Multiples

4.1 Divisibility and Natural Numbers

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.

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.

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.

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.

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

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 prime-power representation of n. 600 = 2 • 2 • 2 • 3 • 5 • 5 = 23 • 31 • 52

PRIMES The Number of Primes There are infinitely many primes.

Tests for Divisibility 4.2 Tests for Divisibility 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.

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.

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.

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.

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.

TESTS FOR DIVISIBILITY Is 81,165 divisible by 3 and by 9? Sum of digits: 8 + 1 +1 + 6 + 5 = 21 81,165 is divisible by 3 since 21 is divisible by 3. 81,165 is not divisible by 9 since 21 is not divisible by 9.

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.

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.

Greatest Common Divisors and Least Common Multiples 4.3 Greatest Common Divisors and Least Common Multiples 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

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. Find the intersection of these sets.

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).

FINDING THE GCD: PRIME FACTORIZATION METHOD Compute the greatest common divisor of Rewrite the numbers as follows:

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

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. Find the intersection of these sets.

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.

Example Finding the LCM: Prime-Power Method Compute the least common multiple of Rewrite the numbers as follows: