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How Do You Find the Greatest Common Factor If There are No Common Factors?

Find the GCF of 8, 10, and 3

Summary

  1. GCF stands for Greatest Common Factor
  2. The greatest common factor is the largest factor a set of numbers has in common
  3. Use prime factorization to break each number down into its prime factors
  4. Pick out the common factors in the factorizations of the three numbers
  5. Since our numbers don't share any common factors other than 1, the GCF is 1

Notes

    1. Prime factorization breaks each number down into its prime factors
    2. Factors are the numbers you multiply together to get another number
    3. A prime number is a number whose only factors are 1 and itself
    1. Factors are the numbers you multiply together to get another number
    2. A prime number is a number whose only factors are 1 and itself
    1. So 2 and 5 are both factors of 10
    2. Notice these are both prime numbers - their only factors are 1 and themselves
    1. The prime factors of 10 are 2 and 5
    1. 2 is a prime number, but 4 is not - 4 has factors other than 1 and itself
    1. We have one 2 that we got at first when we factored 8 into 2•4
    2. Then the other two 2's come from factoring the 4 into 2•2
    3. Notice that if we multiply 2•2•2 we get 8
    1. 3 has no factors other than 1 and itself
    1. 10 factors into 2•5
    2. 8 factors into 2•2•2
    3. 3 doesn't factor any further, so its only prime factor is itself, 3
    1. To find the greatest common factor, we need to see which factors our numbers have in common
    1. This is where all the prime factors are
    1. 2 is in the factorizations for both 10 and 8
    1. There is one 2 in the factorization of 10, but it's already matched up with the first 2 in the factorization of 8
    2. If the other 2's were going to be common factors as well, there would need to be more 2's for 10
    1. There are no 3's in either of the other factorizations
    1. The greatest common factor is determined by the factors all three numbers have in common
    1. So a factor would have to appear in ALL THREE factorizations to be a common factor
    1. There is no 2 in the factorization of 3
    2. So 2 is NOT a common factor
    1. 1 is a factor of every number
    2. So since these numbers don't share any other factors, 1 is the only common factor
    1. You can multiply any number times 1 to get itself
    2. This is called the multiplicative identity
    1. Anything times 1 is just itself