Factors and Multiples

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Factors and Multiples
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Definition of Factors and Multiples

• If one number is a factor of a second number or
  divides the second (as 3 is a factor of 12), then
  the second number is a multiple of the first (as
  12 is a multiple of 3).

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Definition of Factors and Multiples

• Also, if a and b are whole numbers and a does not
  equal 0, then a is a factor of b if and only if
  there is a whole number c such that ac b.

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Components of Factors and Multiples

• Divisibility Tests
• Prime Numbers
• Greatest Common Factor
• Least Common Multiple

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Divisibility Tests

• A quick way to see if numbers are divisible

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Divisibility by 2 or 5

• A number is divisible by 2 if the number
  represented by the units digit is divisible by 2.
  The number is divisible by 2 if the units digit
  is 0, 2, 4, 6, or 8.
• A number is divisible by 5 if the number
  represented by the units digit is divisible by 5.
  A number is divisible by 5 if its unit digit is
  0 or 5.

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Divisibility by 3 or 9

• A number is divisible by 3 if the sum of its
  digits is divisible by 3.
• A number is divisible by 9 if the sum of its
  digits is divisible by 9.

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Divisibility by 4 or 6

• A number is divisible if the number represented
  by the last two digits of a number is divisible
  by 4, then the original number will be divisible
  by 4.
• A number is divisible by both 2 or 3, then it is
  divisible by 6. If it is not divisible by 2 and
  3, then it is not divisible by 6.

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Prime Numbers
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Definition of a Prime Number

• A number which is divisible by 1 and itself, only
  has 2 factors.
• The first 10 prime numbers are 2, 3, 5, 7, 11,
  13, 17, 19, 23, 29.

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Prime Number Test

• Suppose n is a whole number and k is the smallest
  whole number such that k k is greater than n.
  If there is no prime number less than k that is a
  factor of n, then n is a prime number.

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Greatest Common Factor
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Definition of Greatest Common Factor

• For any two nonzero whole numbers a and b, the
  greatest common factor, written GCF(a, b) is the
  greatest factor (divisor) of both a and b.

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How to find Greatest Common Factor

• To find the GCF of 24 and 36
• List all factors of 24 1, 2, 3, 4, 6, 8, 12,
  and 24.
• List all factors of 36 1, 2, 3, 4, 6, 9, 12,
  18, and 36.
• The common factors are 1, 2, 3, 4, 6, and 12.
  The greatest (largest) is 12, therefore 12 is the
  greatest common factor of 24 and 36.

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Least Common Multiple
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Definition of Least Common Multiple

• A number is called a common multiple of two
  numbers if it is a multiple of both.
• Common multiples of 5 and 7 are 35, 70, 105,
  140, 175, etc.
• Because 35 is the smallest common multiple it is
  known as the least common multiple.
• In other words, for any two nonzero whole numbers
  a and b, the least common multiple written LCM(a,
  b), is the smallest multiple of both a and b.

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How to find Least Common Multiple

• For positive integers a and b, LCM(a,b)
• (a b)/(GCF(a,b))
• In other words, multiply the two numbers you are
  finding the least common multiple of and divide
  that answer by the greatest common factor of the
  two numbers.
• Also, when GCF(a, b) 1, LCM(a, b) a b.

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Find LCM (28, 44)

• 28 44 1232
• GCF (28, 44) 4
• 1232/4 308
• 308 is the Least Common Multiple of 28 and 44.

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Now go back to the lesson home page

• When doing the assignment, you can use this
  presentation to help guide you through.
• When doing the quiz, use nothing but your pencil
  and calculator, do not use this presentation.