Chapter 15 - Factors

Most of us remember, from our school days, the word "factors." Chances are you have not encountered the word or the process since. Instead of considering them merely an exercise for students, however, we will show how they can short-cut many problems in multiplication and division.

A factor means, basically, a maker or doer. The word has many applications in English. In mathematics it means one of two or more numbers which, multiplied together, produce the number in question.

6 has two factors: 2 and 3. 2 and 3 are factors of 6 because 2x3 gives 6.

Almost three-quarters of all numbers are factorable. That is, they can be broken down into two or more other numbers which, multiplied together, produce the number you started with. Of the first hundred numbers (from 1 to 100) only 26 are prime numbers. Prime numbers are those that cannot be factored.

1 and 2 are both prime numbers, because they cannot be factored. It is true that 1 x 1 is 1, but we do not consider 1 to be a legitimate factor. It would not be of any use to us in short-cut mathematics, in any event. 3 is also prime. But 4 can be factored into 2 and 2, because 2 x 2 is 4.

Before going into the ways of factoring numbers, let us show the exciting possibilities in their use. They are a powerful short cut because they can save major steps in multiplying and dividing.

To multiply by a factorable number, multiply first by one of its factors and then multiply the result by the other. Where is the short cut? Watch:

In order to use factors, we first find a number that can be factored. Even though in real-life situations you will look at both parts of a multiplication rather than arbitrarily decide that one of them is the multiplier, it is usually quicker to consider the shorter of the two numbers the multiplier.

In this case, 56 is the multiplier. Can it be factored? Can you think of two other numbers that, multiplied together, produce 56? Your knowledge of the multiplication tables should snap the factors 7 and 8 into your mind.

The factor short cut in multiplying any number by 56, then, is to multiply first by 7, then the result of that multiplication by 8. Compare the two ways:

Usual way                                Factor way

The two examples may look about equally time-consuming. But note than in the usual way you multiply first by 5, then by 6, then add the two products to get your final answer. In the factor method you still multiply by two digits—7 and then 8—but you never add any partial products at all. You save roughly one-third the work.

Let's do another before you try one on your own. Check this problem for factor possibilities:

If you have "seen" the factors of 28 at a glance, let us compare methods again:

Usual way                                Factor way

Once more, we managed to skip entirely the step of adding two lines of partial products. Multiply by 4, then by 7, and you have the final answer.

Try this one by yourself. Cover the answer with your pad as you work:
 
This problem has two different, equally correct factor solutions. You can factor 36 into 6 and 6, or into 4 and 9. Any series of accurate factors will produce the same result. Compare the working with these two sets:

Incidentally, there are several other factors of 36. You could use 2 and 18, or 3 and 12. But each of these sets involves two-digit factors. These are useful in longer problems, but it always pays to seek the simplest solution. For simplicity, the choice between two digits and one digit is plain.

How to Factor

We set aside until now the question of factoring itself, so we could show how it works in multiplication. With this specific encouragement, we will get down to the process of factoring before going on to division.

For numbers up to 100, you should be able to recognize factors pretty much at a glance. The most useful factors are single digits, and these can carry you up to 81. Some two-digit numbers can be factored only with three factors or factors of which one has two digits (we will get into the handling of these later), but 74 out of the first 100 numbers can be factored.

Just for a taste, go through the numbers 40 to 49 to see what the possibilities are:

Just because you can factor seven out of these ten numbers (several of them in more than one way) you should not think that you should always use factors. Rule number one for all short cuts remains: look for every possibility, then do it the easiest way. Sometimes you will use factors, and sometimes you will pass them up even if they would be possible, because another short cut happens to be easier or because your new simplified arithmetic is easiest of all in this particular case.

Sharpen your factor-eye by trying the numbers from 30 to 40. Jot down all the possibilities you see on your pad before checking with the following table.

Here are the factors for numbers in the 30's:

So much for the numbers up to 100. Some of those that can be factored are easier to use straight than with factors (38, for instance), but such two-digit factors lead us into higher numbers where they can be very valuable indeed.

It becomes more difficult to recognize factors at sight when we get above 100. Yet factors are even more useful for numbers going into several digits, because they often become dramatic short cuts for bigger numbers.

How would you know, for instance, that 261 can be factored into 9 and 29? Or 536 into 8 and 67?

There are very definite keys developed over the years that show almost at a glance whether a number can be factored with a single digit as one of the factors.

You already know one of them from your work in casting out nines. The digit sum of 261 above is 0; this means that the remainder after dividing by 9 is 0. Obviously, then, 9 must be a factor of 261.

Casting out elevens, too, is merely testing a number for even divisibility by eleven. If there is no elevens-remainder, then eleven is a factor of that number.

Here, in numerical order, are the keys to determining the divisibility of any number by 2 through 12—except for 7. There is a key for 7, but it is so hideously complicated that it is in no sense a short cut.

Key for Divisibility

2 If it is even (the last digit can be divided by 2).

3 If the digit sum is divisible by 3. Just cast out the 9's, and if the remainder is 3 or 6 the number is ex-actly divisible by 3. (If 9 is a factor, it can obviously be broken down further into 3X3, but there isn't much point in doing so because this would raise the other factor in the same proportion.)

4 If the last two digits are divisible by 4. 536 above has 4 as a factor, because 36 can be evenly divided by 4. Two O's as the last two digits also make it divisible by 4.

5 If the last digit is 0 or 5.

6 If it is divisible by both 2 and 3, as outlined above.

7 Too complex a key to be useful here.

8 If the last three digits are divisible by 8. There is a simpler approach to this in actual working, however, since the other factor (when found) will often show you how to increase 4 to 8. Wait and see in the examples to come.

9 If the digit sum is 0.

10 If it ends in 0, of course.

11 If the 1 l's-remainder is 0. Check back with the chap ter on the back-up check.

12 If it is divisible by both 3 and 4, as outlined above.

Try these keys on the two examples mentioned earlier. How can you tell that 9 is a factor of 261? Because the digit sum is 0. How do you know what the other factor is? Simply by dividing with 9, which is simple with a one-digit divider. 9 goes into 261 exactly 29 times, so the factors of 261 are 9 and 29.
 
Now take 536. You know at sight that 4 is a factor, because the last two digits (36) are divisible by 4. The next step is to find the other factor by dividing 536 by 4. This gives you 134 as the other factor. BUT—since 134 is even, you can double the 4 (to 8) and cut 134 in half, to 67. This is the simpler approach mentioned in the key table to divisibility by 8. If you start with 4 and find that the other factor is even, double the 4 to 8 and cut the other factor in half.

As a general rule, it is helpful to use the largest one-digit factor you can, because this cuts the other factor down to the smallest possible size. For 536, it is obviously easier to deal with 8 and 67 than with 4 and 134.

For a bit of practice, factor these numbers. Use your pad to cover the answers:

114      603      392

345      159      139

486      546      243

Warning: one of these numbers is prime, but only one. All the others can be factored.

Here is how we factor all but the next-to-last of the above numbers:

114: It is even, so 2 is a factor. The digit sum is 6, so 3 is also a factor. If both 2 and 3 are factors, then we know 6 is a factor. The factors of 114 are 6 and 19.

345: This ends in 5, so 5 is a factor. 5 into 345 gives you 69 as the other factor. You will note that 3 is also a factor, but this would give you the set 3 and 115.5 and 69 is an easier pair.

486: Digit sum, 0. The factors are 9 and 54.

603: Digit sum, 0. The factors are 9 and 67.

159: Digit sum, 6. 3 is a factor. It is not even, so 2 is not a factor, and if 2 is not a factor then neither is 6. 3 is the largest single-digit factor, and the other (by division at sight) is 53.

546: The digit sum is 6, and it is also even. Both 2 and 3 are factors, so the highest factor is 6. 546 is produced by the factors 6 and 91.
 
392: 2 is a factor (the number is even), but 3 is not because the digit sum is 5. The last two digits, however, are divisible by 4. So we start with the factors 4 and 98. Since 98 is even, we simplify matters by doubling the 4 and halving the 98, and get the factors 8 and 49.

139: This is a prime number. It has no factors. Try all the keys.
243: The digit sum is 0. The factors are 9 and 27.

How to Factor Factors

So far, you have learned to multiply by two one-digit factors. In order to multiply by 56, you multiplied first by 7 and then by 8. In our last number above, however, is it really simpler to multiply by 9 and then by 27 rather than by 243?

It might not seem to be at first thought, but it really is. When you multiply by 243 you have three lines of partial products to add. Using the factors, you have only two (from the two-digit factor).

Often, however, you can factor the factors. You recognize 27 as 3 x 9. So the factors of 243 are 9, 9, and 3.

Extend the factor solution now to include three factors. Instead of multiplying by 243, multiply first by 9. Multiply this result by 9. Multiply this result, in turn, by 3. The answer will be correct, as this comparison of all three solutions shows:

examples. In the usual way you multiply by each of three digits, Do not be deceived by the lines of type occupied by these

Usual way        Two factors      Three factors

then add three lines of partial products. With two factors, you again multiply by three digits, but add only two lines of partial products. With three factors (each of one digit) you multiply still by three digits—but never do any adding at all. See if you can factor the multiplier in the following example into three single-digit factors, and work out the problem on your pad before reading on:

There are several ways you might have factored 336. You could have started with 6 (digit sum 3, number is even) or 4 (36 is divisible by 4). Suppose you started with 4. This would produce the factors 4 and 84. Since 84 is even, you would immediately change the factors to 8 and 42. Since you recognize 42 as the product of 6 and 7, you have the three factors 8, 6, and 7.

One technique we have not yet mentioned, which is frequently quickest, is to start multiplying with your largest factor. It is usually easier for most of us to multiply quickly by smaller digits than by larger ones, and the working figures you multiply get larger at each step.

Our three-factor solution to this problem then goes like this:

Does this answer check out? Try nines- or elevens-remainders on it and see.

Dividing with Factors

Factors are as useful for division as they are for multiplication. Division is by nature the most difficult of the four basic processes for most of us, and you may like the application of factors to division most of all because they frequently permit us to divide with single digits rather than more complex numbers.

Division is just the reverse of multiplication, so the use of factors in division is just the reverse of their use in multiplication. The technique is to factor the divider if you can, then divide by each of the factors. Each division, of course, is into the result of the last division rather than into the original number divided.

Which way, even at first glance, looks easier? The faster nature of the factor short cut becomes even more dramatic if you divide the second factor into the result of the first division without bothering to rewrite, like this:

The work should be clear. You started at the bottom and worked up, setting up the second division into the answer of the first. It is merely a condensed picture of the two stages shown separately in the first explanation, and is the way you would actually do it in practice—assuming you did not merely jot down the answer to the first division without bothering to rewrite the problem.

Which Factor to Use First

In multiplication, we start with the largest factor and work down. In division, we usually start with the smallest factor and work up—for precisely the same reason, in reverse. The easiest digits to divide by are usually the smallest, and our division stages get smaller as we progress. So for the earlier divisions into longer numbers, it is most often easier to start with the smallest factors.

Watch out for special cases, however, particularly in problems with remainders—which so many in division have. In these cases you may be able to get through the first division without a remainder if you handle it properly. This simplifies things.

In general, match the factor used first to the number divided. If one factor is odd and the other even, start by dividing with the odd factor if the number divided is odd and by dividing with the even factor if the number divided is even.

Here is an example that illustrates this:

The first step is to factor the divider into 4 and 7. Second, note that this division cannot come out even; it must have a remainder. You know this because even into odd can never produce an even answer (although odd into even can). So, in this case, we start with the odd factor rather than the even one:

 If we started with the even factor, here is what the first step would look like:

Obviously, the other approach is easier to begin with. Dividing this result by the second factor, now, we produce the final answer:

 This illustration does not bother to put the decimal point and zeros into the second number divided because you do not need to either. Just keep mentally "bringing down" zeros after the decimal point.

If you try dividing 2654.75 by 7, you will get the same final answer. But it is more work. You had a remainder on the first division by 4, so you have to divide through two remainders instead of just one.

Matching odd and even will not always avoid this, but it often will. When you cannot avoid a remainder in the first result, by the way, carry it only to as many decimal places as you will need in the final answer. There is no point to dividing on and on with a remainder that may never come out even.

One other key to watch for in picking your first-division factor is to see if either of the factors of your divider is also a factor of the number divided. If it is, you know that the first division by this factor must come out even.

Now try one division by factors on your own:

Cover the answer with your pad until you have finished.

The factors of 72 are recognizable at sight: 8 and 9. Since the number divided is even, we will start with the even factor, then divide that result by the other factor. The illustration will be in condensed style. See if your working agrees with this:

Division with Three Factors

Just as it often pays to use three factors in multiplying, so you can use three factors in division to speed up and simplify your work.

In multiplying, you use each of the three (or more) factors in turn to avoid adding lines of partial products. In dividing, you use each of the factors in turn to avoid the extra complications of dividing with a number of two or more digits —where you can. When one of the factors has to be of two or more digits, you will still find the division simpler than dividing with a still longer number.

Suppose we factor this problem:

No matter how we tackle it, this is admittedly one of the divisions most of us hate to face.

First, see if the divider can be factored. 567 has a digit sum of 0, so we know at once that 9 is a factor. 9 into 567 (without writing down the problem, of course) gives us the other factor, 63. 63 we recognize as 7 x 9. So 567 has three one-digit factors: 9, 9, 7.

Since all the factors are odd, you may as well start with the smallest. Stack your working as we did before, and the factor solution looks like this: 

What factoring really accomplished in this case, as you can see, was to reduce the solution to three divisions of single digits each, rather than one division by a number of three digits.
 
It is time now to try a three-factor division yourself. Cover the answer with your pad until you have finished this problem:

The first step is to see if the divider can be factored. The digit sum is 8, so it is not divisible by 9 or 3. It is even, so it is divisible by 2, but moreover the last two digits are divisible by 4, so 4 is a factor. We start by factoring it into 4 and 56. Since 56 is even, we double the 4 and cut the 56 in half: factors, 8 and 28.

We prefer to work with single-digit factors if we can, so we further factor the 28 to 4 and 7. All the factors we need for 224, then, are 4, 7, and 8.

Since the number divided is even, let's start with the 4 and work up:

That's all there is to it. That rather fearsome division problem is reduced, thanks to factoring, to three quick and simple single-digit divisions.

Sometimes you can factor a number into one one-digit factor and one two-digit factor, but cannot further factor the two-digit factor at all. You may still save time by using these two factors, though, just as you can in multiplication. Dividing by a two-digit number is so much easier than dividing by a three-digit number (even in the shorthand method) that it will probably pay you to use the factors—especially since you have already gone to the trouble of factoring the divider.
Every time you use factors, you will become fonder of them. They are the third major area of short-cut conversions, following naturally after breakdown and aliquots.

Now we will take up the fourth type of conversion.

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