Chapter 6 - No-Carry Multiplication

Multiplying, according to the same estimates we mentioned before, averages about 20% of the figuring done in normal business.

But while it is used less than addition, multiplication is dreaded by more people, and done poorly or inaccurately by more people, than either addition or subtraction.

Perhaps this is because multiplication, particularly of one long number by another long number, becomes so fearfully complicated in comparison to the simpler process of adding or subtracting. Most of us have little trouble visualizing that adding ten lines of numbers may involve more work, but is really no more complex, than adding two lines of digits. But multiplying 2,958 by 165 is something that few of us can really "see" as a whole. We tackle it step by step, by pure rote, in inefficient and inherently slow traditional methods.

Our new way to multiply solves a large part of this. It involves three secrets. Two of them you already know from adding and subtracting. The third is brand-new.

The first secret is the one we use throughout this book: work from left to right. Tackle 165 as 165, not as 5, 6, 1. The less we have to use methods that violate the plain common sense of the way we normally read numbers, the better off we are. Our number sense becomes sharper instead of becoming dulled by backward absurdities.

Working from left to right also makes our new method of multiplying a self-estimating system, just as our left-to-right addition or subtraction is.

There is no simple way to work from left to right in the classical method of multiplying. But our no-carry method works just as easily from left to right as from right to left, so you will find it natural to work in the proper direction.

The second secret, again, is the same as its equivalent secret in adding or subtracting: "see" only the answer, combining digits at a glance. This is simply a matter of practice, but chances are you have already had more practice at this than you had for adding. Most schools spend far more time on drill in multiplication tables than they do on drill in addition and subtraction tables. So you are probably closer to mastering this step in multiplication than you were for the two earlier processes.

If you were taught in the standard way, however, you would do well to begin practicing the deletion of the slowdown steps taught in school. Instead of reading the example 4 x 7 as "4 times 7 is 28," make a conscious effort to look at it and think only "28." You do not look at "me" and think "m and e is 'me.'"

In fact, the entirely new way to multiply involved in the third step will bring up quite a different way of looking at 4 x 7. You will never, oddly enough, think the whole product at all, but only half of it at a time.

The third secret is the new method. It is radically different from the traditional way to multiply because you never have to "carry." The greatest trouble with standard multiplication, and the greatest source of errors, is carrying. It is very much like the difficulty in "borrowing" in subtraction. Either you forget to carry, or carry twice, or carry the wrong figure—and wind up hating numbers.

The no-carry method of multiplying works without remembering to carry at all. It may look a little strange at first, but once you try it a few times you will get the idea.
 
The easiest way to approach this method is to take apart a sample multiplication and see what makes it tick. Make sure you fully understand every step of this, because once you understand why the system works as it does you will find it very easy to use. If you simply try to learn the technique by rote, however, it will always seem complicated.

Let's take this multiplication apart:

4 7
x 8
5 6
3 2
3 7 6

Look at the answer, digit by digit, and see how it really develops.

The first digit, 3, is simply the left-hand (tens) digit of 4 times 8—32.

Look back at the example and note this. The first digit of this answer is merely the tens, or left-hand, digit produced by multiplying the first digit of the number multiplied by the multiplier.

The second digit is a little more complicated. This 7 is the sum of two other digits. It is the sum of the right-hand (units) digit of the multiplication we just examined—4 times 8—and the left-hand (tens) digit of 7 times 8. The right-hand digit of 4 times 8 is 32, or 2. The left-hand digit of 7 times 8 is 56, or 5. 2 plus 5 is 7—the middle digit in our answer.

Look back at the example again to make sure this is completely clear. Read the above explanation again if you need to.

If you remember our earlier comments about left-to-right working, in which we pointed out that each digit increases in value by a factor of ten as it moves one place to the left, then you can see why the middle digit in this answer is the sum of the unit part of the 4 times 8, and the tens part of the 7 times 8. It is because the 4 in 47 is really ten times 4 because of its position—or 40.
 
The last digit in this answer is 6. This is simply the right-hand (units) digit of 7 times 8—56.

This is a new way of looking at multiplication for most people. Get it clear now, and everything that follows will fall into place naturally and easily.

Now let us try multiplying those same numbers left to right in the new method, using the understanding above of how the answer really develops. If the method seems unclear at any point, re-check the explanation above.

4 7 x 8

Step one: Look at 4 x 8 to see only what the left-hand (tens) digit of the product will be. In other words, is 4 x 8 in the teens, twenties, thirties, forties, or what?

4 x 8 is in the 30's. The tens digit of this pair is 3. For the moment, you do not care what the right-hand, or units, digit is. All you care about is the 3.

For the first digit of your answer, put down that 3:

3 Step two: Now look at 4 x 8 to see what the right-hand, or units, digit of this pair is ... what the full product of 4 X 8 "ends in." The units digit is 2, the 2 of 32. Remember that 2 for just an instant while you look at 7 x 8 to see what its left-hand, or tens, digit will be. 7 x 8 is in the 50's. Add this 5 to the remembered 2 and put down the total as the second digit of your answer. 2 plus 5 is 7, so your answer now looks like this:
3 7 Step three: Look at 7 x 8 again to see only what digit the product ends in. The right-hand, or units, digit of 7 x 8 is 6—the 6 of 56. Put it down as the last digit in your answer:

3 7 6

Pause here for a moment to let this sink in. It is just as shocking an idea in its own way as is the idea of complements for adding and subtracting, and just as useful. But, as with complements, you need a bit of time to adjust to the thought.

There is one point in step two when you must remember one digit while "seeing" another one to add to it. Check the traditional process taught in school, however, and you will find that you had to juggle three digits at this point. You had to carry the 5 from 56 while noting the 32, then remember the 3 from 32 while adding the 2 and 5 and putting down 7. After that, you had to remember to put down the 3 from 32. The new no-carry method is at least one-third simpler—and produces the answer from left to right as well.

Here is another run-through to reinforce your grasp of this method:

8 3 x 9

Step one: 8 x 9 is in the 70's. Write down 7 as the first digit of your answer:

7 Step two: 8X9 ends in 2. Remember 2. 3 x 9 is in the 20's. Add the remembered 2 and the 2 from the 20's and put down 4:

7 4 Step three: 3X9 ends in 7. Put down 7 as the last digit in your answer:

7 4 7

If any element along the way does not seem to make sense, go through the three steps again with pencil and pad. This is really an incredibly simple idea, but it is vastly different from the way we were taught to work with numbers.

Now we will try one more, adding another digit. This means simply that we shall do step two twice. More properly, steps "one" and "three" are special steps for the extreme left and right digits of the number multiplied. Step "two" is the step done for every pair of digits across the number multiplied; once for a two-digit number, twice for three digits, and so on.

Here is how it works with a three-digit number:

5 3 2
x 7

One: 5 x 7 is in the 30's. Put down 3: 3

Two (1): 5X7 ends in 5. Remember 5. 3 x 7 is in the 20's. Add 5 and 2. Put down 7:

3 7

Two (2): 3X7 ends in 1. Remember 1. 2 x 7 is in the 10's (teens). Add 1 and 1. Put down 2:

3 7 2

Three: 2X7 ends in 4. Put down 4:

3 7 2 4

That is the basic system. It is that simple, and that revolutionary. If there had been twenty digits in the number multiplied, you would simply have repeated step two until you got to the end.

Get out your pad, open to a clean page, and go through the steps exactly as described for the following example. Do not try it on other random problems yet, however, because there are two special techniques for special cases yet to be revealed.

4 7
X 4

After you have done this, check your answer by the usual method of multiplying. If it checks out, good. If not, go back through the steps and see where you went wrong.

If it still does not come out right, compare your working with this description of the proper steps:

Step one: 4 x 4 is in the 10's. Put down 1. step two: 4X4 ends in 6. 7 x 4 is in the 20's. Add 6 and 2. Put down 8.

Step three: 7x4 ends in 8. Put down 8. The final answer is 188.

Now for the two special cases. Both are important, because examples involving them will crop up repeatedly in your work with numbers.

How to Handle Zeros

Sometimes, in going through the no-carry multiplying system, you will match a pair of digits whose product is less than ten. It might be 3 x 2. This product is 6. There is no left-hand, or tens, digit at all. In effect this product is in the "zeros."

For this system, however, you must use a left-hand digit. Otherwise the answer will not come out right. So no-carry multiplication always depends on using a left-hand digit even if that digit is zero.

When you come across 3x2, you will consider it in the zeros, just as 3 x 4 is in the 10's, and 3 x 7 is in the 20's.

The reason for keeping this in mind is that your left-hand and right-hand product digits are essential to keeping your imaginary "carries" in proper order. Later, when we come to working with two or more digits in the multiplier, you will find them important for keeping your columns in line too. This is really no more difficult than remembering to put down the zero in 5 x 6 when working from right to left, and performs basically the same function.

Suppose, for instance, you faced this example:

5 1 4
x 7

Step one: 5 x 7 is in the 30's:

3

Step two (1): 5X7 ends in 5. 1 x 7 is in the zeros. 5 plus zero is 5:

3 5

Step two (2): 1X7 ends in 7. 4 x 7 is in the 20's. 7 plus 2 is 9:

3 5 9

Step three: 4x7 ends in 8:

3 5 9 8

One other important point about products whose left-hand, or tens, digits are in the zeros should be kept in mind. Get in the habit of putting down a zero as the first digit of the answer if this is what the problem produces. It is not essential for one-line answers such as those in the above examples, but it is absolutely essential to getting two-line answers lined up properly.

This is what I mean:

1 6
x 4

Step one: 1 x 4 is in the zeros. Put down 0:

Step two: 1X4 ends in 4. 6 x 4 is in the 20's. Add 4 and 2. Put down 6:

0 6

Step three: 6x4 ends in 4. Put down 4:

0 6 4

Your answer is merely 64. The zero in front of it does not change its value. But when you come to multiplying by numbers of two or more digits, you will see the necessity of this technique. It is for precisely the same reason, as we said a page back, that in the traditional method you put down the zero of thirty or forty at the right of the answer in a two-line multiplication.

But since this is a new way of doing things, be sure to get into the habit of doing it this way whenever the problem works out like this. Try it on these two samples. Use your pad:

4 9       3 8 6
x 2       x 2

Be sure to do these. Simply reading through practice examples, intending to do them later, will not teach you how to do speed mathematics. Theory and practice go hand in hand.

Check your results and the steps you went through in the two samples above against this explanation:

First sample. Step one: 4 x 2 is in the zeros. Put down 0.
 
Step two: 4X2 ends in 8. 9 x 2 is in the 10's. 8 plus 1 is 9. Step three: 9x2 ends in 8. Answer: 098.

Second sample. Step one: 3 x 2 is in the zeros. Put down 0. Step one (1): 3 x 2 ends in 6. 8 x 2 is in the 10's. 6 plus 1 is 7. Step two (2): 8 x 2 ends in 6. 6 x 2 is in the 10's. 6 plus 1 is 7. Step three: 6x2 ends in 2. Answer: 0772.

If you neglected to put down the zeros in front of these answers, do (for the sake of your swift mastery of two-digit multipliers) go back and do them properly now. Simple repetition, pencil in hand, means a great deal in getting accustomed to new techniques such as these.

The new way of looking at half-products may come a little hard at first. You were taught to think "6 times 8 is 48." Now, in two separate steps, you are learning to look at 6 times 8 and (for one step) think only "40's," then (for another step) think only "8." Don't worry about that part yet. It is really quite a simplification of the multiplication tables, and there is some practice ahead to help give you the knack.

Before going on to the final step in no-carry multiplying, get a firmer grip on the steps so far by doing these two examples. Turn to a clean page of your pad and try your teeth on these:

9 3 6    7 4 9
x 4       x 6

We will not go through the explanation of these in detail. Do them thoughtfully and carefully, working at this point for full understanding and accuracy rather than speed. Speed will follow because you are now working in what is essentially a simpler and more logical manner.

Once you have done the two examples, check your results by repeating the two problems according to your old method. If the answers check out, fine. If not, study the steps in detail to find out where you went wrong.

Now for the final step.

Recording Tens

So far, all our examples have been carefully selected to avoid one special situation that is really more complicated in he traditional method than it is in no-carry multiplying. But the situation does need a technique to handle it, and we have a simple and automatic one.

This example will demonstrate the special situation. Go through it step by step and find the new problem:

8 9
x 6

Step one: 8 x 6 is in the 40's:

4

Step two: 8x6 ends in 8. 9 x 6 is in the 50's. 8 plus 5 is—
STOP! You cannot put down a single digit standing for the sum of 8 and 5. This goes over ten. In our new way to add, we do not even try to add them. Instead, we subtract the complement of 8 (2) from 5 and put down 3:

4 3

But this is not quite right. When we use a complement, we must also record a ten. How do we record a ten here?

One of the secrets of this simplified mathematics is that we let the tens take care of themselves. We record them in adding, or cancel them in subtracting. We never, never try to remember them. That would be inefficient.

In multiplying, then, we will simply use the same written symbol we use in adding. We will underline. In this case, an underline will raise the value of the underlined digit by one— just as, in subtracting, a slash reduces the value of the slashed digit by one. Since the underline is in effect carried back from the 3 (the underline represents the 1 in 13, which is one place to the left), we will underline the digit one place to the left.

So our answer now looks like this:

4 3

Step three: 9x6 ends in 4:

4 3 4

Since the underline raises the value of the underlined digit by one, we read our answer like this:

5 3 4

Is this correct? Check the problem and see. Just as important, or even more important, see if the logic of it is clear.

If this seems the least bit complicated, review in your mind the schoolbook approach to this problem. Here is the thinking you were instructed to do: "6 x 9 is 54. Put down the 4, carry the 5. 6 x 8 is 48. We carried a 5. 8 plus 5 is 13. Put down the 3. Carry one from the 13. Add the 1 of the 13 to the 4 of 48. 1 plus 4 is 5. Put down 5." Which, once you are equally familiar with both approaches, is really simpler?

Let's go through this new process once more in detail:
4 6 8
x 3

Step one: 4 x 3 is in the 10's: 1

Step two (1): 4 x 3 ends in 2. 6 x 3 is in the 10's. 2 plus 1 is 3:

1 3

Step two (2): 6 x 3 ends in 8. 8 x 3 is in the 20's. 8 and 2 are complements. Zero, record:

1 3 0

Step three: 8x3 ends in 4:

1 3 0 4

Until you are thoroughly accustomed to reading slashed and underlined digits accurately without hesitation, it is good practice to rewrite such answers:

1 3 0 4 becomes
1 4 0 4

Now try doing a problem that involves this situation on your own. Write your answer, left to right, before going on. Then check your steps against the explanation that follows:

7 8
x 8

Here is the way the no-carry method works on this problem:

Step one: 7 x 8 is in the 50's:

5

Step two: 7X8 ends in 6. 8 x 8 is in the 60's. Complement of 6 (4) from 6 is 2, and record the ten:

5 2

Step three: 8x8 ends in 4:

5 2 4

Rewrite this answer as 624, and you are done.

Sometimes, too, your recorded ten will affect a first-place zero. Here is such a case: 1 9x 8

Work this one out yourself, being sure to put down a zero if the left-hand, or tens, digit of any of the products is in the zeros, and see if this changes as you go through the full answer.

Work it out now on your pad.

If you did each step correctly, your answer should look like 052, which you rewrite as 152.

Do the next two problems on your pad. Be sure to go through exactly the steps we have been demonstrating, and check your answers to make sure they are right. If you hesitate too much, or come up with a wrong answer on two or three tries, then a review of the last few pages is in order.

9 3 6    4 8 5
x 7       x 6

How Complements Help

By this time, you have surely noticed a surprising and delightful fact: complement addition is a tremendous aid to no-carry multiplication. You need have no worry about remembering when to record a ten. The occasion is signaled to you automatically. You record a ten every time you use a complement or add to ten, and that is all. One goes with the other.

Whenever you use a complement, of course, this also gives you the digit to enter in the answer more quickly and accurately than would trying to add (say) 9 plus 9 and getting 18—of which you would have to put down the 8 and carry the ten. Working the new way, you think simply "8, record."

It is almost foolproof, once it becomes a habit. No-carry multiplication is the closest possible approach to the secret of the soroban: to make as much of the operation as possible mechanical, so you can give your attention to the digit-by-digit sums and products and divisions without worrying about carrying and holding numbers in your mind. Since you are released from much of the mental labor of ordinary mathematics, you can concentrate on the single most important skill needed to handle this quickly and easily: your ability to "see" 6x7 as "40's" and "ends in 2" without hesitation or effort. The next chapter will go more fully into this.

Two-Digit Multipliers

No part of no-carry multiplication changes when we approach multipliers of two or more digits. It would be theoretically possible to produce the answer to two- or three-digit multipliers in one operation, but this means juggling four digits in your mind at once. This, for most of us, is impractical. Some "short-cut" mathematics books do urge this method, but it is really going in precisely the wrong direction for true speed. Unless years of practice go into them, the methods for producing the answer to 59 times 38 in one operation, on one line, are more apt to get mixed up and give a wrong answer than to speed up your results.
 
You now know how to produce a left-to-right answer to any multiplication by one digit with greater speed and accuracy, as well as ease. Let us stick to this head start, and do longer problems in the simplest and fastest way. We will use a different line for each digit in the multiplier. We will arrange these lines, however, in the reverse of the classical system. Start your top line with the left digit of the multiplier, and put each following line one place to the right.

This method is both easier and faster once you get the hang of it. It keeps you working from left to right—which is more natural, and also makes the system self-estimating.
 
Step one: 2 x 9 is in the 10's:
 
Step two: 2X9 ends in 8. 6 x 9 is in the 50's. Complement of 8 (2) from 5, and record:
 
Step three: 6x9 ends in 4:

Watch this demonstration:

Step one: 2 x 9 is in the 10's:
 
1

Step two: 2X9 ends in 8. 6 x 9 is in the 50's. Complement of 8 (2) from 5, and record:

Step three: 6x9 ends in 4:

Now, for the second line. This we get by multiplying the 8 by each digit in turn of the number multiplied, and we place the answer one place to the right (we work always from left to right):

Step one: 2 x 8 is in the 10's:

Step two: 2X8 ends in 6. 6 x 8 is in the 40's. 6 and 4 are complements. Zero, record:

Step three: 6x8 ends in 8:

Now you simply add these lines from left to right. The easiest way to handle the recorded tens is to add each underline as one, rather than rewrite the two lines before adding:

This should be clear through every step. If not, recheck your understanding of the preceding pages.

There is one special point to watch carefully. You recall the stress we put on putting down a first digit for each line, even if that first digit happens to be a zero. If you forget to do this, your lines for each digit in the multiplier will get out of order and your answer will be wildly wrong. Notice how it works in this example:

If you had not put down the zero in front of the 7 in that first line, you might very possibly have lined up the two partial-products like this:
 

Not only is the answer absurdly wrong, but it is the type of error that is infernally hard to catch. You might do the problem over several times, get every line right, and still get the wrong answer. So watch, very carefully, your first-place zeros.

Go through the process once yourself. Doing is the secret of remembering:

Cover the solution below with your pad while you work out this example.

If you remembered that you always put down a left-hand digit even if it is a zero, then your answer looked like this:

Note how we handled the two recorded tens in adding the lines of partial answers. You are already learning to read underlined digits as increased in value by one. Occasionally, in very complex multiplications with three, four, or more digits in the multiplier you will need to record more than one ten for the same place. That is, sometimes you will need to add two or three to a digit already put down as you work from left to right.

In any multiplication likely to become this involved, you may wish to handle the addition of partial answers just as if it were a major addition problem, and record tens on your fingers to be noted beneath each digit to the left. But multiplication seldom becomes this complex. In most cases, you will find it quicker and easier to use two or even three underlines when you need to.

Now it is time to handle a couple of two-digit multiplications entirely on your own. Cover up the answers below with your pad while you use it to work out the answers. Use your new techniques exclusively.

After you have finished, compare your working with these answers:

Larger Multipliers

When you get into three- and four-digit multipliers, you simply follow the steps you already know. They are no different in kind, only in degree. Just add one more line for each new digit in the multiplier, stepping one place to the right for each line, and remember to put down a left-hand digit even if it is a zero in order to keep the lines in proper order.

There is no magic to getting the right answer for a long multiplication problem. There are two ways of getting very rapid, quite accurate estimates—the slide rule, and the self-estimating feature of no-carry multiplication—but for a full, complete answer you simply have to go patiently through all the steps. Those steps are made more natural and easier as well as quicker by this method.

Watch the step-by-step development of the answer to the following example. You should be able to understand why each new digit appears without trouble. If anything does not seem clear, review the last few pages and try again. It might be a good idea to follow with pencil and pad, too.

Now add, from left to right of course:

Go through the mental processes in your mind, making sure that you too would put down each new digit as it appears in the step-by-step unfolding of this answer. Note especially how the underlines are handled.

Now try a couple of three-digit multiplier problems on your own. Work from left to right, in the no-carry method, and remember to put down a left-hand digit for your first product in each line even if it is a zero:

You will find the detailed working of these two examples at the end of this chapter. Get your pad now, though, and go through them to the end before reading on. Save your working for a check against the solutions to come later.

Automatic Estimating

One of the beautiful features of left-to-right, no-carry multiplication is the way it produces quick estimates. It is as fully automatic in this respect as is left-to-right addition or subtraction.

There is no easy and accurate way of doing this with traditional multiplication. Yet it is built right in, at no extra cost, to any left-to-right system.

You can get a two-digit estimate in a twinkling. You can get a three-digit estimate (which equals the accuracy of almost any slide rule) while the man with the slide rule is still getting out his "slip stick" and setting it.

This is not a criticism of the slide rule. If you must do a great deal of multiplying and dividing and are satisfied with rounded-off answers—which the slide rule provides by its very nature—then it is well worthwhile getting one and learning how to use it. It is not hard. But do not pass up this estimating short cut even if you have a slide rule, because the system is both useful and impressive. It also works when your slide rule is somewhere else.
The technique for estimating with no-carry multiplication to any required degree of accuracy is simply this: Multiply as far as you have to and stop. Raise the last digit by one for each two digits in the multiplier.

Suppose you face a really formidable multiplication such as the cost of 53,926 items at $48.75 each. You must give a rapid approximation to three digits.

All you need to do is quickly scrawl each part of your new no-carry multiplication as far as three digits from the left. Here is how you do it:

So far, you have 260. There are four digits in the multiplier—count 4, 8, 7, 5—so raise the 0 by 2. Now you have 262.

A slide rule would not do any better. Carry the multiplication further, if you wish, and see how close we are.

Does this mean $262,000 or $2,620,000? One simple rule gives you an unfailing answer to this question. Your answer has exactly as many digits as the total of the two numbers multiplied. Just add the digits in these two numbers, and figure on that many in the answer. Special note: If the first digit of the answer is a zero (the first digit of the first line of partial answers), this must be counted too.

In the above estimate, your answer is $2,610,000. Try working it out and see—keeping in mind that two of the digits in the multiplier are behind the decimal point and therefore are a fraction.

An estimate of would have five digits in the final answer, or 68,900. The total number of digits in number multiplied and multiplier is six, but in the answer one of that total is lost in the initial zero.

Two other special points are interesting in this matter of estimating. First, note that you work your answer out to only as many places as you need and, in order to do it, you work out each partial answer to this same number of places starting at the top left. For a three-digit estimate, you will have three digits in the top partial answer, two in the second, and one in the third. If the first line begins with a zero (as in the one above) then you will go to four digits in the first line. Should your multiplier have twenty digits in it, you would ignore all but the first few.

Perhaps you wonder why you raise the last digit by one for each two digits in the multiplier. Check back to the section on estimating in the chapter on subtraction, and you will find a very similar rule. The reason is this: The average of any random number of digits including 0 is 4½. The average for each two lines in addition is therefore 9—plus the likelihood of tens recorded (or carried back to this column) from the column to the right, at the rate of about one for each 2½ lines. The best average for estimating, then, is to increase your final digit by one for each two lines in the addition. And the number of lines in the final addition of a multiplication problem is determined by the number of digits in the multiplier: one line of partial answer for each digit.

So raise the final digit of your estimate by one for each two digits in the multiplier. Forget any extra digits, and count five as two, seven as three.
 
Practice estimating these two problems accurate to three digits. Use pencil and paper. Remember to raise the last digit of your estimate in the way described above, and to count the digits in both numbers and use this total as the number of digits in your answer—including an initial zero if it appears in the top line of partial answers.

Do these now:

The estimates of these appear at the end of the chapter.

Do them yourself, though, before you look.

• • •

Here are the answers to the two three-digit problems you were asked to work out on page 78. Compare them with your solutions:
 
And here is the way we estimate to three-digit accuracy the two examples at the top of this page:

Add 2(4 digits in multiplier): 4 19 - Note especially the two underlines, meaning two recorded

Nine digits:

The next chapter will help you to develop greater familiarity and speed with these techniques. If you feel that everything in this chapter is completely clear, go on ahead. If not— review.

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