Number Theory: Prime & Composite Numbers 5.1 Number Theory: Prime & Composite Numbers

Objectives Determine divisibility. Write the prime factorization of a composite number. Find the greatest common divisor of two numbers. Solve problems using the greatest common divisor. Find the least common multiple of two numbers. Solve problems using the least common multiple.

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 Natural numbers that are multiplied together are called the factors of the resulting product.

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.

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 number 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.

Example 2: Prime Factorization using a Factor Tree Example: Find the prime factorization of 700.

Greatest Common Divisor To find the greatest common divisor of two or more numbers, Write the prime factorization of each number. Select each prime factor with the smallest exponent that is common to each of the prime factorizations. Form the product of the numbers from step 2. The greatest common divisor is the product of these factors. 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.

Example 3: Finding the Greatest Common Divisor Example: Find the greatest common divisor of 216 and 234.

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: Write the prime factorization of each number. Select every prime factor that occurs, raised to the greatest power to which it occurs, in these factorizations. Form the product of the numbers from step 2. The least common multiple is the product of these factors.

Example 5: Finding the Least Common Multiple Example: Find the least common multiple of 144 and 300.

Example 6: Solving a Problem Using the Least Common Multiple 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?

The Integers; Order of Operations 5.2 12

Graph integers on a number line. Use symbols < and >. Objectives Define the integers. Graph integers on a number line. Use symbols < and >. Find the absolute value of an integer. Perform operations with integers. Use the order of operations agreement. 13 13

Define the Integers The set consisting of the natural numbers, 0, and the negatives of the natural numbers is called the set of integers. Notice the term positive integers is another name for the natural numbers. The positive integers can be written in two ways: Use a “+” sign. For example, +4 is “positive four”. Do not write any sign. For example, 4 is also “positive four”. 14 14

Example 2: Using the Symbols < and > Insert either < or > in the shaded area between the integers to make each statement true: 4 3 1 5 5 2 0 3 15 15

Absolute Value The absolute value of an integer a, denoted by |a|, is the distance from 0 to a on the number line. Because absolute value describes a distance, it is never negative. Example: Find the absolute value: |3| b. |5| c. |0| Solution: 16 16

Additive Inverses Additive inverses have the same absolute value, but lie on opposite sides of zero on the number line. When we add additive inverses, the sum is equal to zero. For example: 18 + (18) = 0 ( 7) + 7 = 0 In general, the sum of any integer and its additive inverse is 0: a + (a) = 0 17 17

Example 6: Evaluating Exponential Notation Evaluate: a. (6)2 b. 62 c. (5)3 d. (2)4 Solution: 18 18

Order of Operations Perform all operations within grouping symbols. Evaluate all exponential expressions. Do all the multiplications and divisions in the order in which they occur, working from left to right. Finally, do all additions and subtractions in the order in which they occur, working from left to right. 19 19

Example 7: Using the Order of Operations Simplify 62 – 24 ÷ 22 · 3 + 1. 20 20