(FYI, this lesson covers the California State Math Standard 1 for Number Sense, sub-standard 2.4, which says that you will "determine the least common multiple and the greatest common divisor of whole numbers...")
To start us off, let's consider this question:
Normally you would use what we've been calling in class the "arrow method", where you would write the two main numbers followed by arrows, after which you would write their factors in order from least to greatest, like so:
Next you would circle the largest factor that both sets of numbers have in common. In this case, the greatest common factor for 16 and 24 is 8:
There is another way to find the GCF, and though in this example (using 16 and 24 once more) it might seem like more work is being used to find the answer, you'll really appreciate using this method with much bigger numbers!
The first step you take is to use factor trees to find the prime factors of both 16 and 24. We've been using factor trees a lot lately, so this part shouldn't be difficult for you:
After you've determined that 16's prime factors are 2 • 2 • 2 • 2 and that 24's prime factors are 2 • 2 • 2 • 3, you'll want to write those two rows of factors one above the other, then circle the factors that both 16 and 24 have in common:
At this point you're going to ignore the prime factors that 16 and 24 don't have in common. Take the common prime factors (in this case three sets of 2) and multiply the 2s one time for each red circle:
You can see that when we compare the answer we just got with the answer we received using the first method, both answers are the same: the GCF of 16 and 24 is 8. Like I said earlier, you'll find great success using this method when finding the GCF for really big numbers!
Now it's your turn to practice. I'd like you to choose 10 pairs of numbers: the first five pairs can be relatively easy ones (ranging from between 10 and 50), while the last five pairs need to be a bit more challenging for you (definitely above 50 and as high as you'd like to go, within reason!). Work out the GCF for all ten of those number pairs, record your answers, then check your work at this cool website: Greatest Common Factor Tool. The site allows you to enter in two numbers and it will automatically tell you what their GCF is. I'm trusting that you'll do this AFTER you've worked out the problems on your own using the factor tree method you just learned!
If you'd like to challenge yourself, find 10 more sets of numbers, but those sets should have three numbers each (the factor tool website will let you find the GCF of three numbers, too!)
