Lesson 1: Factors and Multiples of Whole Numbers

Todays Objectives Students will be able to demonstrate an understanding of factors of whole numbers by determining the: prime factors, greatest common factor (GCF), least common multiple (LCM), square root, cube root Including Determine the prime factors of a whole number Explain why the numbers 0 and 1 have no prime factors Determine, using a variety of methods, the GCF, or LCM of a set of whole numbers, and explain the process

Number Sets Natural Numbers (N) Whole Numbers (W) Integers (I or Z) Numbers are divided up into several different sets: Natural Numbers (N) 1,2,3,4,5,… Often called the “counting” numbers Whole Numbers (W) 0,1,2,3,4,5,… Includes all of the natural number set Integers (I or Z) …,-3,-2,-1,0,1,2,3,… or can be written as 0, ±1, ±2, ±3,… Includes all of the natural and whole number sets Rational Numbers (Q) Any number that can be written as a ratio (fraction) Terminating decimal numbers, repeating decimal numbers Includes all of the natural, whole, and integer number sets Irrational Numbers (Q’) Any non-terminating, non-repeating decimal number Examples: 𝜋, ℮ Real Numbers(ℝ) All of the above number sets

Summary of Number Sets Real Numbers Irrational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers

Factors of Whole Numbers A whole number is a number that is a member of the set W:{0, 1, 2, 3, 4 ,5,….}. Notice that 0 is a whole number, but it is not a member of the set of natural numbers, N: {1, 2, 3, 4, 5,….}. Factors of a number are numbers that multiply together to make that number. For example: 6 and 4 are factors of 24 (6 x 4 = 24). 24 has the following factors: 1, 2, 3, 4, 6, 8, 12, 24

Prime Numbers and Factors Any whole number greater than one that has only two distinct factors (one, and itself) is called a prime number Example: 13 is prime because it’s only factors are 1 and 13. Numbers greater than 1 that are not prime are called composite numbers. When the factors of a number are also prime, they are called prime factors. Example: 12 has prime factors 2 x 2 x 3. We call this the prime factorization of 12.

1 and 0 The number 1 is not a prime number because it is not divisible by any whole numbers other than itself The number 0 is not prime because it does not have 2 distinct factors

Examples List the whole number factors of 40 Solution: The whole number factors of 40 are the whole numbers by which 40 is divisible. They are 1, 2, 4, 5, 8, 10, 20, 40. List the prime factors of 40. Solution: The prime factors are the prime numbers by which 40 is divisible. They are 2 and 5. Write the prime factorization of 40. Solution: Writing a number as a product of other prime numbers is called the prime factorization 40 = 2 x 20 (2 is prime, 20 is not) 40 = 2 x 2 x 10 (2 is prime, 10 is not) 40 = 2 x 2 x 2 x 5 (all factors are now prime)

Example (You do) List the first 10 prime numbers  Solution: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 For the number 72, list the whole number factors, prime factors and write the prime factorization Solution Factors: 1,2,3,4,6,8,9,12,18,24,36,72 Prime Factors: 2,3 Prime Factorization: 2 x 2 x 2 x 3 x 3 One method of writing a prime factorization is to use a factor tree

Factor trees A factor tree is a diagram used to write the prime factorization of a prime number

Greatest Common Factor (GCF) The greatest common factor of two or more whole numbers is the largest whole number that is a factor of two or more numbers. Example: The GCF of 12 and 18 is 6. 12 has factors 1, 2, 3, 4, 6, 12 18 has factors 1, 2, 3, 6, 9, 18 The largest shared factor between these two number is 6, so 6 is the GCF. Here are some techniques to finding the GCF of different numbers.

Example Determine the GCF of 24, 72, and 180  Solution 1: List all the factors of the three numbers; choose the largest factor shared by all three. 24 has factors 1,2,3,4,6,8,12,24 72 has factors 1,2,3,4,6,8,9,12,18,24,36,72 180 has factors 1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180 As we can see, the GCF is 12.

Example Solution 2: Factor the numbers into products of powers of prime factors. The GCF is the product of the common powers with the smallest exponents associated with each power. Prime factorization of 24 is 2 x 2 x 2 x 3 = 23 x 31 Prime factorization of 72 is 2 x 2 x 2 x 3 x 3 = 23 x 32 Prime factorization of 180 is 2 x 2 x 3 x 3 x 5 = 22 x 32 x 51 The bases 2 and 3 are common to all three prime factorizations. Since the power with base 2 with the smallest exponent is 22 and the power with a base of 3 with the smallest exponent is 31, the GCF is 22 x 31 = 12.

Example Solution 3: Divide the numbers by common prime factors until all the quotients do not have a common prime factor. In the method shown, the quotients are written under the dividends (divided numbers). The numbers 2, 6, and 15 do not have a common prime factor. The product of the common prime factors is the GCF. This is the product of the divisors in the left column: 2, 2, and 3. Thus, the GCF is 2 x 2 x 3 = 12. 2 24 72 180 12 36 90 3 6 18 45 15

Example (You do) Determine the GCF of 48, 80, and 120 Solution: 2 48 80 120 2 24 40 60 2 12 20 30 6 10 15 The GCF is 2 x 2 x 2 = 8.

Least Common Multiple (LCM) The least common multiple of two or more whole numbers is the smallest whole number that is a multiple of two or more whole numbers. For example, the least common multiple of 12 and 18 is 36 because it is the smallest whole number that is a multiple of both 12 and 18. Multiples of 12 = 12, 24, 36, 48, 60, 72,…. Multiples of 18 = 18, 36, 54, 72, ….. There are a few techniques to finding the LCM, as shown in the next example.

Example Determine the LCM of 8, 12, and 30. Solution 1: List the multiples of 8, 12, and 30 until a common multiple is found Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, 120 Multiples of 12 = 12,24, 36, 48, 60, 72, 84, 96, 108, 120 Multiples of 30 = 30, 60, 90, 120 The smallest number that is a multiple of 8, 12, and 30 is 120, so the LCM is 120.

Example Solution 2: Factor each number into products of powers of prime factors. The LCM is the product of the common powers that have the largest exponents associated with them, along with any non-common powers. Prime factorization of 8 = 2 x 2 x 2 = 23 Prime factorization of 12 = 2 x 2 x 3 = 22 x 31 Prime factorization of 30 = 2 x 3 x 5 = 21 x 31 x 51 The LCM is the product of the common powers with the largest exponents (23 and 31), along with the non-common power (51). Thus, the LCM is 23 x 31 x 51 = 120.

Example Solution 3: use a similar division technique as we used for the GCF. This time, we keep dividing until none of the numbers in a row have a common prime factor. (Bring 2 down from the previous row) The LCM is the product of the left column and the bottom row. LCM = 2 x 2 x 3 x 2 x 5 = 120. 2 8 12 30 4 6 15 3 1 5

Example (You do) Determine the LCM of 16, 18, and 20. Solution: 2 16 9 10 4 5 Bring down the 9 LCM = 2 x 2 x 4 x 9 x 5 = 720

Homework Pg. 140-141 # 3-8ace, 9, 11, 13, 17, 20 Read: Section 3.2: Perfect Squares, Perfect Cubes, and their roots