5.1 Number Theory 213 Suppose you decide to start your own candy company. You would like to sell boxes containing 48 pieces of candy. You must decide how to arrange the pieces within the box. You could have six rows with eight pieces in each row, or four rows with twelve pieces in each row, or three layers each with four rows with four pieces in each row. Each of these possibilities involves writing the number 48 as the product of two or more smaller numbers. In this section, we will discuss many other similar problems related to the writing of numbers as the product of smaller numbers. Number Theory SECTION 5.1 LEARNING GOALS Upon completion of this section, you will be able to: 7 Classify natural numbers as prime or composite. 7 Understand the rules of divisibility for the numbers 2, 3, 4, 5, 6, 8, 9, and 10. 7 Write the prime factorization of a natural number. 7 Determine the greatest common divisor of a group of natural numbers. 7 Determine the least common multiple of a group of natural numbers. 7 Understand the nature of prime numbers. Why This Is Important Writing numbers as the product of smaller numbers has many applications to the branch of mathematics we know as number theory. For example, banks and retail companies use number theory to secure online transactions. This chapter introduces number theory, the study of numbers and their properties. The numbers we use to count are called the counting numbers or natural numbers. Because we begin counting with the number 1, the set of natural numbers begins with 1. The set of natural numbers is frequently denoted by N: N {1, 2, 3, 4, 5, } = … Any natural number can be expressed as a product of two or more natural numbers. For example, 8 2 4, 16 4 4, = × = × and 19 1 19. = × The natural numbers that are multiplied together are called factors of the product. For example, 2 4 8 × = Factors A natural number may have many factors. For example, what pairs of numbers have a product of 18? ⋅ = ⋅ = ⋅ = 1 18 18 2 9 18 3 6 18 The numbers 1, 2, 3, 6, 9, and 18 are all factors of 18. Each of these numbers divides 18 without a remainder. If a and b are natural numbers, we say that a is a divisor of b or a divides b, symbolized a b, | if the quotient of b divided by a has a remainder of 0. If a divides b, then b is divisible by a. For example, 4 divides 12, symbolized 4 12, | since the quotient of 12 divided by 4 has a remainder of 0. Note that 12 is divisible by 4. The notation |7 12 means that 7 does not divide 12. Note that every factor of a natural number is also a divisor of the natural number. Caution: Do not confuse the symbols a b| and a b/ ; a b| means “a divides b” and a b/ means “a divided by b” a b ( ). ÷ The symbols a b/ and a b ÷ indicate that the operation of division is to be performed, and b may or may not be a divisor of a. Prime and Composite Numbers Every natural number greater than 1 can be classified as either a prime number or a composite number. Ronstik/Shutterstock
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