Speed Mathematics Simplified (49 page)

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

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.

Watch how it works in this case:

The factors of 63 are 7 and 9. Divide first by 7:

Now divide the
result
by the other factor, 9:

Compare this solution to the usual method of working:

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.