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Factors & Multiples Copyright©2001 Lynda Greene Greatest Common Factor

Introduction to Factors What are they and how do I find them?

Example: Say you want to find the factors of 8. The factors of 8 are all the numbers that will divide into 8 evenly. (In other words, they do not have a remainder) So take the numbers from 1 to 8 and divide 8 by each of them. 1) 8 8) 8 2) 83) 84) 8 5) 86) 87) 8 What are factors and how do I find them? 8 42 remainder 1 If you get an answer with a remainder, the number you divided by is not a factor of 8, so cross out these answers. Now let’s look at the numbers that are left

1) 88) 82) 84) The numbers on the left are the factors of 8, factors: 1, 2, 4 and 8 But, did you notice that the numbers on the top and the numbers you divided by (on the left) are the same? That’s because we are finding the factors two at a time, The number on the left and the number on top are both factors of 8. So to save time we don’t have to divide by every number from 1 to 8, we can go halfway and stop.

How do we know when we have gotten halfway and can stop? 1) Write the number with two little branches below it 8 2) Starting with ‘1 x 8’ Write all the pairs of factors that divide evenly into 8 1 x 8 2 x 4 4 x 2 8 x 1 3) This is where they start to repeat, STOP HERE! You don’t need to write these repeating numbers down If you write the factors of the number using the following system, you can see where your stopping point will be. All the factors of 8 are right here in this little box.

Practice: Find the factors of the following numbers x 12 2 x 6 3 x 4 1 x 32 2 x 16 4 x 8 1 x 81 3 x 27 9 x 9 1 x 48 2 x 24 3 x 16 4 x 12 5 : remainder 6 x 8 7 : remainder Here’s where the numbers start to repeat 4 x 3, etc. so stop here. Factors of 12: 1, 2, 3, 4, 6, 12 5 : remainder 6 : remainder 7 : remainder 8 x 4(repeat) Make sure you check all the numbers up to the number on the bottom right, this is where they start to repeat. Factors of 32: 1, 2, 4, 8, 16, 32 Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 Factors of 81: 1, 3, 9, 27, 81 You can stop here since there are no more numbers between these two factors on the bottom We can stop checking numbers as soon as we reach this number Stop, since the next number is 8 Read the factors in this order Down the left side Up the right side

Practice Problems: (Hit enter to see the answers) Find all the possible Factors of the following numbers 1) 48 5) 64 2) 78 6) 75 3) 59 7) 110 4) 32 8) 83 Answers: 1) 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 2) 1, 2, 3, 6, 13, 26, 39, 78 3) 1, 59 (prime number) 4) 1, 2, 4, 8, 16, 32 5) 1, 2, 4, 8, 16, 32, 64 6) 1, 3, 5, 15, 25, 75 7) 1, 2, 5, 10, 11, 22, 55, 110 8) 1, 83 (prime)

Now that we know what the factors of a number are and how to find them, we are ready to learn how to find the Greatest Common Factor.

Greatest Common Factor (GCF)

We want: Greatest: The biggest number Common: all the terms have in common Factor: from a list of their factors In other words: 1) Find all the factors 2) Pick the biggest factor they all have in common (this is the GCF) x 12 2 x 6 3 x 4 1 x 6 2 x 3 1 x 9 3 x 3 Greatest Common Factor (GCF) The name Greatest Common Factor tells us what to do to the terms and also what we are looking for in the numbers GCF = 3

Find the GCF for these numbers: x 12 2 x 6 3 x 4 1 x 30 2 x 15 3 x 10 5 x 6 1 x 24 2 x 12 3 x 8 4 x 6 1 x 18 2 x 9 3 x 6 Notice that these numbers also have 1, 2 and 3 in common, but the Greatest Common Factor is the BIGGEST number they have in common, which is a ‘6’. GCF = 6

Practice Problems: (Hit enter to see the answers) Find the GCF for each set of numbers 1) 24, 18, 48 5) 26, 78, 117 2) 12, 33, 78 6) 75, 200, 350 3) 14, 35, 105 7) 33, 77, 110 4) 32, 64, 128 8) 27, 81, 135 Answers: 1) 6 2) 3 3) 7 4) 16 5) 13 6) 25 7) 11 8) 9

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