Thinking Mathematically

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Thinking Mathematically
Number Theory: Prime and Composite Numbers

The Set of Natural Numbers

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. All three statements are symbolized by writing b|a.

Discuss the Rules of Divisibility [p. 194]
Discuss using a calculator to check divisibility

Prime Numbers A prime number is a natural number greater than 1 that has only itself and 1 as factors.

Composite Numbers 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 (if the order of the factors is disregarded). For example 700 can be written in the following way. 700 = 2 x 2 x 5 x 5 x 7 The prime factors of 700 are 2, 5, and 7.

“Factor Trees” The prime factors of a natural number can be found by constructing a “factor tree.” Write the given number as a product and continue to factor each composite number until only prime numbers remain. The “prime factorization” of 40 is determined by the prime numbers at the bottom of each branch of the tree. 40 x 8 5 4 2 x 40 = 2 x 2 x 2 x 5 = 23 x 5 2 x The prime factors of 40 are 2 and 5.

Finding the Greatest Common Divisor of Two or More Numbers Using Prime Factorization
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.

Example of Finding the Greatest Common Divisor
40 x 8 5 4 2 24 x 6 4 3 2 x 2 x 40 = 2 x 2 x 2 x 5 24 = 2 x 2 x 2 x 3 The greatest common divisor of 24 and 40 is 2 x 2 x 2 = 8.

Finding the Least Common Multiple Using Prime Factorization
To find the least common multiple 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 of the Least Common Multiple
40 x 8 5 4 2 12 x 3 4 2 40 = 2 x 2 x 2 x 5 12 = 2 x 2 x 3 The least common multiple of 12 and 40 is 2 x 2 x 2 x 3 x 5 = 120.

Thinking Mathematically
Number Theory: Prime and Composite Numbers