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1.1 Factors and Multiples of Whole Numbers

Prime numbers = # that can only be divided by 1 and itself

Examples:

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When a factor of a number has exactly two divisors –1 and itself, the factor is a prime factor.

However, when can express ALL numbers in terms of prime factors. We call this prime factorization and write a number as a product of its prime factors.

The factors of 12 are 1, 2, 3, 4, 6, and 12.

12 = 2 . 2 . 3 or 22 . 3

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There are two methods to find the prime factorization of a number:

Method 1:Factor Trees

Example 1: Write the prime factorization of 2646.

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Method 2:Use repeated division.STEP 1: Begin by dividing the number by the least (smallest) prime factor. STEP 2: Divide by this prime factor until it is no longer a factor.STEP 3: Choose another prime factor and continue with steps 1 and 2.STEP 4: Stop when the quotient reaches 1.

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Now you try!Write the prime factorization of 3150.

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For 2 or more natural numbers we can determine their greatest common factor, the greatest factor that all the numbers have in common.

Example 2: Determine the GCF of 126 and 144.

Method 1:Determine all the factors for each number.

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Method 2: Find all the factors of one number and check which numbers are divisible by the second.

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Method 3:Use prime factorization.STEP 1: Find the prime factors of each number using prime factorization.STEP 2: Determine which prime factors appear in both numbers.STEP 3: Multiple the repeated numbers to find the GCF.

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Now you try!Find the GCF of 234 and 342.

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What are the multiples of 22? To generate multiples simply multiply a number by the natural numbers (1, 2, 3, 4...)

For 2 or more natural numbers we can determine their least common multiple, the smallest number that is divisible by both numbers.

Example 3: Determine the LCM of 14, 20, and 42.

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Method 1: List all the multiples of each number until the same number appears in all 3 lists.

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Method 2: List the multiples of the largest number and check to see which numbers are divisible by the other numbers.

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Method 3: Use prime factorization.STEP 1: Find the prime factors of all of the numbers using prime factorization.STEP 2: Determine the greatest power of all the prime numbers used in any list.STEP 3: Multiply the greatest power of each prime factor to get the LCM.

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Now you try!Find the LCM of 28, 42, and 63.