Section 5.1 Number Theory: Prime & Composite Numbers Objectives 1. Determine divisibility. 2. Write the prime factorization of a composite number. 3. Find the greatest common divisor of two numbers. 4. Solve problems using the greatest common divisor. 5. Find the least common multiple of two numbers. 6. Solve problems using the least common multiple. 9/16/2011 Section 5.1 1
Number Theory and Divisibility Number theory is primarily concerned with the properties of numbers used for counting, namely 1, 2, 3, 4, 5, and so on. The set of natural numbers is given by N = { 1,2,3,4,5,6,7,8,9,10,11,... } The natural numbers that are multiplied are called the factors of the product. 9/16/2011 Section 5.1 2
Number Theory and Divisibility If a and b are natural numbers, a is divisible by b if the operation of dividing a by b leaves a remainder of 0. This is the same as saying that b is a divisor of a, or b divides a. This is symbolized by writing b a. Example: We write 12 24 because 12 divides 24 or 24 divided by 12 leaves a remainder of 0. Thus, 24 is divisible by 12. Example: If we write 13 24, this means 13 divides 24 or 24 divided by 13 leaves a remainder of 0. But this is not true, thus, 13 24. 9/16/2011 Section 5.1 3
Prime Factorization A prime number is a natural number greater than 1 that has only itself and 1 as factors. A composite number is a natural greater than 1 that is divisible by a number other than itself and 1. The Fundamental Theorem of Arithmetic Every composite number can be expressed as a product of prime numbers in one and only one way. One method used to find the prime factorization of a composite number is called a factor tree. 9/16/2011 Section 5.1 4
Prime Factorization Prime Factorization using a Factor Tree Example: Find the prime factorization of 700. Solution: Start with any two numbers whose product is 700, such as 7 and 100. Continue factoring the composite number, branching until the end of each branch contains a prime number. 9/16/2011 Section 5.1 5
Prime Factorization Prime Factorization using a Factor Tree Example Continued Thus, the prime factorization of 700 is 700 = 7 x 2 x 2 x 5 x 5 = 7 x 2 2 x 5 2 = 2 2 5 2 7 Notice, we rewrite the prime factorization using a dot to indicate multiplication, and arranging the factors from least to greatest. 9/16/2011 Section 5.1 6
Greatest Common Divisor Pairs of numbers that have 1 as their greatest common divisor are called relatively prime. For example, the greatest common divisor of 5 and 26 is 1. Thus, 5 and 26 are relatively prime. To find the greatest common divisor of two or more numbers, 1. Write the prime factorization of each number. 2. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. 3. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. 9/16/2011 Section 5.1 7
Greatest Common Divisor Finding the Greatest Common Divisor Example: Find the greatest common divisor of 216 and 234. Solution: Step 1. Write the prime factorization of each number. 9/16/2011 Section 5.1 8
Greatest Common Divisor Example Continued The factor tree at the left indicates that 216 = 2 3 x 3 3. The factor tree at the right indicates that 234 = 2 x 3 2 x 13. Step 2. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. Which exponent is appropriate for 2 and 3? We choose the smallest exponent; for 2 we take 2 1, for 3 we take 3 2. 9/16/2011 Section 5.1 9
Greatest Common Divisor Example Continued Step 3. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. Greatest common divisor = 2 x 3 2 = 2 x 9 = 18. Thus, the greatest common factor for 216 and 234 is 18. 9/16/2011 Section 5.1 10
Least Common Multiple The least common multiple of two or more natural numbers is the smallest natural number that is divisible by all of the numbers. To find the least common multiple using prime factorization of two or more numbers: 1. Write the prime factorization of each number. 2. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. 3. Form the product of the numbers from step 2. The least common multiple is the product of these factors. 9/16/2011 Section 5.1 11
Least Common Multiple Finding the Least Common Multiple Example: Find the least common multiple of 144 and 300. Solution: Step 1. Write the prime factorization of each number. 144 = 2 4 x 3 2 300 = 2 2 x 3 x 5 2 Step 2. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. 144 = 2 4 x 3 2 300 = 2 2 x 3 x 5 2 9/16/2011 Section 5.1 12
Least Common Multiple Example Continued Step 3. Form the product of the numbers from step 2. The least common multiple is the product of these factors. LCM = 2 4 x 3 2 x5 2 = 16 x 9 x 25 = 3600 Hence, the LCM of 144 and 300 is 3600. Thus, the smallest natural number divisible by 144 and 300 is 3600. 9/16/2011 Section 5.1 13
Least Common Multiple Solving a Problem using the LCM Example: A movie theater runs its films continuously. One movie runs for 80 minutes and a second runs for 120 minutes. Both movies begin at 4:00 p.m. When will the movies begin again at the same time? Solution: We are asked to find when the movies will begin again at the same time. Therefore, we are looking for the LCM of 80 and 120. Find the LCM and then add this number of minutes to 4:00 p.m. 9/16/2011 Section 5.1 14
Least Common Multiple Solving a Problem using the LCM Example Continued Begin with the prime factorization of 80 and 120: 80 = 2 4 x 5 120 = 2 3 x 3 x 5 Now select each prime factor, with the greatest exponent from each factorization. LCM = 2 4 x 3 x 5 = 16 x 3 x 5 = 240 Therefore, it will take 240 minutes, or 4 hours, for the movies to begin again at the same time. By adding 4 hours to 4:00 p.m., they will start together again at 8:00 p.m. 9/16/2011 Section 5.1 15
Section 5.7 Arithmetic and Geometric Sequences Objectives 1. Write terms of an arithmetic sequence. 2. Use the formula for the general term of an arithmetic sequence. 3. Write terms of a geometric sequence. 4. Use the formula for the general term of a geometric sequence. 9/16/2011 Section 5.7 1
Sequences A sequence is a list of numbers that are related to each other by a rule. The numbers in the sequence are called its terms. For example, a Fibonacci sequence term takes the sum of the two previous successive terms, i.e., 1+1=2 1+2=3 3+2=5 5+3=8 9/16/2011 Section 5.7 2
Arithmetic Sequences An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the common difference of the sequence. Arithmetic Sequence Common Difference 142, 146, 150, 154, 158, d = 146 142 = 4-5, -2, 1, 4, 7, d = -2 (-5) = -2 + 5 = 3 8, 3, -2, -7, -12, d = 3 8 = -5 9/16/2011 Section 5.7 3
Arithmetic Sequences Write the Terms of an Arithmetic Sequence Example: Write the first six terms of the arithmetic sequence with first term 6 and common difference 4. Solution: The first term is 6. The second term is 6 + 4 = 10. The third term is 10 + 4 = 14, and so on. The first six terms are 6, 10, 14, 18, 22, and 26 9/16/2011 Section 5.7 4
The General Term of an Arithmetic Sequence Consider an arithmetic sequence with first term a 1. Then the first six terms are Using the pattern of the terms results in the following formula for the general term, or the n th term, of an arithmetic sequence: The nth term (general term) of an arithmetic sequence with first term a 1 and common difference d is a n = a 1 + (n 1)d 9/16/2011 Section 5.7 5
The General Term of an Arithmetic Sequence Example: Find the eighth term of the arithmetic sequence whose first term is 4 and whose common difference is -7. Solution: To find the eighth term, a 8, we replace n in the formula with 8, a 1 with 4, and d with -7. a n = a 1 + (n 1)d a 8 = 4 + (8 1)(-7) = 4 + 7(-7) = 4 + (-49) = -45 The eighth term is -45. 9/16/2011 Section 5.7 6
Geometric Sequences A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence. Geometric Sequence 1, 5, 25, 125, 625, 4, 8, 16, 32, 64, 6, -12, 24, -48, 96, Common Ratio r r r = = = 5 1 8 4 12 6 = = = 5 2 2 9/16/2011 Section 5.7 7
Writing the Terms of a Geometric Sequences Example: Write the first six terms of the geometric sequence with first term 6 and common ratio ⅓. Solution: The first term is 6. The second term is 6 ⅓ = 2. The third term is 2 ⅓ = ⅔, and so on. The first six terms are 6,2, 2 3, 2 9, 2 27, and 2 81. 9/16/2011 Section 5.7 8
The General Term of a Sequence Consider a geometric sequence with first term a 1 and common ration r. Then the first six terms are Using the pattern of the terms results in the following formula for the general term, or the n th term, of a geometric sequence: The nth term (general term) of a geometric sequence with first term a 1 and common ratio r is a n = a 1 r n-1 9/16/2011 Section 5.7 9
The General Term of a Sequence Example: Find the eighth term in the geometric sequence whose first term is -4 and whose common ratio is -2. Solution: To find the eighth term, a 8, we replace n in the formula with 8, a 1 with -4, and r with -2. a n = a 1 r n-1 a 8 = -4(-2) 8-1 = -4(-2) 7 = -4(-128) = 512 The eighth term is 512. 9/16/2011 Section 5.7 10