Section 5.1 Number Theory: Prime

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Section 5.1Number 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.

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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
• The natural numbers that are multiplied are
  called the factors of the product.

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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 ba.
• Example We write 1224 because 12 divides 24 or
  24 divided by 12 leaves a remainder of 0. Thus,
  24 is divisible by 12.
• Example If we write 1324, this means 13 divides
  24 or 24 divided by 13 leaves a remainder of 0.
  But this is not true, thus, 1324.

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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.

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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.
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Prime Factorization Prime Factorization using a
Factor TreeExample Continued

• Thus, the prime factorization of 700 is
• 700 7 x 2 x 2 x 5 x 5
• 7 x 22 x 52
  
• Notice, we rewrite the prime factorization using
  a dot to indicate multiplication, and arranging
  the factors from least to greatest.

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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,
• 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.

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Greatest Common DivisorFinding the Greatest
Common Divisor

• Example Find the greatest common divisor of 216
  and 234.
• Solution Step 1. Write the prime factorization
  of each number.

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Greatest Common DivisorExample Continued

• The factor tree at the left indicates that 216
  23 x 33.
• The factor tree at the right indicates that 234
  2 x 32 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 21,
  for 3 we take 32.

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Greatest Common DivisorExample 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 32
  2 x 9 18. Thus, the greatest common factor for
  216 and 234 is 18.

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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.

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Least Common MultipleFinding 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 24 x 32
• 300 22 x 3 x 52
• Step 2. Select every prime factor that occurs,
  raised to the greatest power to which it occurs,
  in these factorizations.
• 144 24 x 32
• 300 22 x 3 x 52

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Least Common MultipleExample Continued

• Step 3. Form the product of the numbers from step
  2. The least common multiple is the product of
  these factors.
• LCM 24 x 32 x 52 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.

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Least Common MultipleSolving 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
  400 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 400
  p.m.

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Least Common MultipleSolving a Problem using the
LCMExample Continued

• Begin with the prime factorization of 80 and 120
• 80 24 x 5
• 120 23 x 3 x 5
• Now select each prime factor, with the greatest
  exponent from each factorization.
• LCM 24 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 400 p.m., they will start
  together again at 800 p.m.