How do you explain the greatest common factor?

Answer 1

Let's think of a factor of a number as some number that divides it. For example, look at the number 12, 2 divides it since 12 divided by 2=6. We say 2 divides it if when we divide 12 by 2 and we have no remainder. Next, 3 divides 12 since 12 divided by 3=4, and 4 divides it since 12 divided by 4 is 3. All these things that divide 12 are called factors of 12. So 6 is a factor of 12 since 12 divided by 6=2 and 12 divided by 12 is 1 so 12 is a factor 1 is always a factor since 12 divided by 1 is 12. So the factors of 12 are: 1,2,3,4,6,12, Now, let's look at another number, say 18. The factors would be 1,2,3,6,9, and 18. All these numbers go into ( divide) 18 with no remainder. Now to find the greatest common factor, we can look at the common factors first. That is to say to find the greatest common factor of 12 and 18 we can list all the common factors. They are; 1,2,3, and 6. Now which one of these is the largest number? It is 6 so that is the GCF or greatest common factor of 12 and 18. If the largest common factor is 1, we say the numbers are relatively prime. This is NOT the same as saying the number is prime, but there is a relation. Relatively prime means the GCF is 1, prime means the only factors are 1 and the number itself. Now, the method I used will always work, but it is tedious. Another method is to factor the numbers into primes and compare the prime factorization of the two numbers. Take the smallest exponent of each common prime. In our example, 12= 2^2x3 and 18 = 3^2x2 The common primes are 2 and 3 and the smallest power of 2 can be found in 18 where the power of 2 is 1 . The smallest power of 3 can be found in 12 where the power of 3 is 1. So to find the GCF we multiply 2x3 and get 6 just as we already found by listing factors and finding the common ones.