BMCC: Math Tutorials: Numbers:
Factoring
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Factors and Products
Finding Factors
In this section we deal with factors and factoring. For the purposes of this discussion, we will restrict ourselves to whole numbers only. A factor is a whole number that can be multiplied by another whole number to produce a particular product. A product is any number that results from multiplying two or more factors.
That's a circular definition, so let's focus on the term product first. Since a product is the result of multiplying two factors together, that means any whole number can be a product, because at the very least you can generate any number n by multiplying 1 times that number n itself.
For instance, the number 3 is the product of 1 times 3. So the numbers 1 and 3 are both factors of the product 3. The number 2 has to be multiplied by 1.5 to produce the product 3, but 1.5 is not a whole number, so 2 is not a factor of 3. No number larger than a particular product can ever be a factor of that product, because a number larger than 3 would have to be multiplied by a value less than 1 to produce the product 3.
Another way to think of this is that any factor of a number will divide that number evenly, with no remainder. Thus, for the product 3, the only factors are 1 and 3.
As another example, the number 6 is the product of 1 times 6. Thus, the numbers 1 and 6 are both factors of the number 6. In this case, though, you can also produce the product 6 by multiplying 2 times 3, so both 2 and 3 are also factors of 6. No other whole numbers can be multiplied to produce the product 6-- if you check to see whether 4 is a factor, you see that you have to multiply 4 by 1.5, which is not a whole number, so 4 is not a factor of 6. Similarly, the number 5 would have to be multiplied by 1.2 to produce the product 6, so 5 is not a factor of 6 either.
So for the product 6, we can say that the factors are 1, 2, 3, and 6.
Suppose there is a product that can be produced by multiplying a specific factor one or more times, such as 4, which can be produced by multiplying 2 times 2. The number 2 meets the criteria for being a factor of 4, but even though it is multiplied by itself, it is still listed only once in the factor list. The factors of 4 are thus 1, 2, and 4. The same principle applies for larger products which may have more repeating factors-- a factor is only listed once in the factor list for a product.
Determining the factors for a product is not an exact procedure. Instead, it's more like a process of elimination. For small numbers, like we showed above, the process is simple. For larger numbers, it's usually fairly easy to find some factors, but finding all of the factors and knowing that you found all of the factors can be more difficult. Basically, the process consists of finding some factors of the product, and breaking down those factors themselves, until you have a list of factors that cannot be broken down any further.
The most complete approach would be to start with the number 1 and divide the product systematically by each increasing number to determine which numbers can be divided into the product evenly (with no remainder). When you reach a point where the factors begin to repeat, you know there are no further factors. For example, consider the product 12. We know that 1 is a factor, because 1 times 12 equals 12. Next we see if 2 is a factor. Since 12 can be divided evenly by 2 with a quotient of 6, then 2 and 6 are also factors. Now we try 3, and since 3 times 4 equals 12, both 3 and 4 are factors. The next number in the progression to try is 4, but we have already determined that 4 is a factor-- that is, the factors are beginning to repeat. That means all of the factors have been found, and it's not necessary to continue trying 5 (which is not a factor), 6 (which we already determined was a factor when we checked 2), etc. We can say that the factors of 12 are 1, 2, 3, 4, 6, and 12.
If you use this progressive approach, you can see that there is no reason to continue examining numbers past the half-value of the product, because for any number greater than the half-value of the product, there is no corresponding whole number by which it can be multiplied to yield the product. If you are looking for factors of 21, for instance, which has a half-value of 10.5, there could never be any factor larger than 10 (in fact, the largest factor of 21 is 7).
Sometimes it's easiest to start with a small factor. No matter how large a number n is, if the number is even (i.e., the last digit of the number is 0, 2, 4, 6, or 8), then 2 will be a factor of n. The number that can be multiplied by 2 to produce n will also be a factor, and may be able to be broken down itself into additional factors.
Two other small numbers that are easy to test as factors are 3 and 5. If the individual digits in a product add up to a sum that is evenly divisible by 3, then 3 is a factor of that product. For example, the sum of the digits in 651 is 12 (6 + 5 + 1 = 12), and 12 is evenly divisible by 3, so 3 is a factor of 651 (in fact, 3 times 217 equals 651). The sum of the digits in 217 is 10 (2 + 1 + 7 = 10), and 10 is not evenly divisible by 3, so 3 is not a factor of 217.
To determine if 5 is a factor of a number, you need only to look at the last digit of the number. If the last digit is either 0 or 5, then the number is evenly divisible by 5, and therefore 5 is a factor.
Next: Prime Numbers
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