Read Speed Mathematics Simplified Online
Authors: Edward Stoddard
Speed Mathematics Simplified (48 page)
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 12âexcept 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
| Â Â 2 | Â | If it is even (the last digit can be divided by 2). |
| Â Â 3 | Â | 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 exactly divisible by 3. (If 9 is a factor, it can obviously be broken down further into 3 Ã 3, but there isn't much point in doing so because this would raise the other factor in the same proportion.) |
| Â Â 4 | Â | 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 0's as the last two digits also make it divisible by 4. |
| Â Â 5 | Â | If the last digit is 0 or 5. |
| Â Â 6 | Â | If it is divisible by both 2 and 3, as outlined above. |
| Â Â 7 | Â | Too complex a key to be useful here. |
| Â Â 8 | Â | 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. |
| Â Â 9 | Â | If the digit sum is 0. |
| 10 | Â | If it ends in 0, of course. |
| 11 | Â | If the 11's-remainder is 0. Check back with the chapter on the back-up check. |
| 12 | Â | 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. BUTâsince 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:
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: | Â | 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: | Â | 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: | Â | Digit sum, 0. The factors are 9 and 54. |
| 603: | Â | Digit sum, 0. The factors are 9 and 67. |
| 159: | Â | 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: | Â | 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: | Â | 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: | Â | This is a prime number. It has no factors. Try all the keys. |
| 243: | Â | 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 Ã 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
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 digitsâbut 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:
