Welcome back to our series on number properties. This article will introduce GMAT quant problems that revolve around the relationships between the digits of multi-digit integers. This topic can be tricky because we all know how the base-10 system works, but this knowledge is so ingrained that we hardly ever think about it. We can easily get confused when asked to manipulate knowledge related to place value.
Problems that revolve around the relationship between the digits of multi-digit integers
This topic is different from other number properties topics in that there’s little or no theory to learn. We will jump right into some official GMAT problems so that you can get familiar with how GMAT quant builds questions around this topic. Here’s a straightforward one as a warmup:
Official GMAT problem for practice
If r and t are three-digit positive integers, is r greater than t?
- The tens digit of r is greater than each of the three digits of t.
- The tens digit of r is less than either of the other two digits of r.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
The question asks which three-digit integer is greater, so we can think of the three digits as a “tiebreak” system. If the integers have the same hundreds digit, then we will need to know about their tens digits. And if they also have the same tens digit, then we will need to know about their units digits. Of course, if r has a greater hundreds digit than t does, the other digits are irrelevant – r is greater.
Statement 1 isn’t useless, but it isn’t sufficient by itself. The tens digit of r is greater than each digit of t. This still leaves room for r to be greater than or less than t, depending on the hundreds digits of each integer.
Statement 2 again provides data about the tens digit of r, this time comparing it to the other digits of r. This obviously can’t be sufficient by itself, since we know nothing at all about t. But when combined with the data from statement 1, this data is useful. If the tens digit of r is less than the hundreds digit of r (the one that matters most in this problem) and greater than any digit of t, then the hundreds digit of r is greater than the hundreds digit of t. This means that r is greater than T, and the statements together are sufficient. The correct answer is C.
Problem about the nature of multi-digit addition
Let’s try a slightly more complex problem:
If x, y, and z are three-digit positive integers and if x = y + z, is the hundreds digit of x equal to the sum of the hundreds digits of y and z?
- The tens digit of x is equal to the sum of the tens digits of y and z.
- The units digit of x is equal to the sum of the units digits of y and z.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
This is an interesting problem about the nature of multi-digit addition. The question is built around whether a digit in a given place of the sum is equal to the sum of the digits occupying the same place in the addends. An addend is a number being added with another number. Let’s look at a couple of example three-digit addition problems:
426 375 + 351 + 592 777 967
Apex does not recommend doing addition like this on GMAT quant, because it is much faster and more efficient to be able to perform such calculations mentally, without the use of your whiteboard or scratch paper. But in this case, it’s helpful to see the “columns” formed by the units, tens, and hundreds places. In the first example, each digit in the sum is equal to the sum of the corresponding digits in the addends. This is not true in the second example. The controlling element is whether the two digits in the addends have a sum greater than 9. If they do, as in the tens place of the second example, it is necessary to “carry” a 1 to the next digit place to the left. Elementary, but perhaps difficult to think about so many years after learning to add.
Now we can consider the statements. Statement 1 tells us that the tens digit of the sum x is indeed equal to the sum of the tens digits of the addends y and z. Since this is true, there will be no “carrying” to alter the values in the hundreds place. Therefore the hundreds digit of the sum x must be equal to the sum of the hundreds digits of y and z. Statement 1 is sufficient.
Statement 2 is qualitatively similar to statement 1 but tells us instead about the units digits. Interestingly, the data provided by statement 2 is already proven by statement 1. Since the tens digit of the sum x is equal to the tens digits of the addends y and z, there must not have been any “carrying” from the units place to the tens place to alter the values in the tens place. The units digit of the sum x must represent the full combined values of the units digits of the addends y and z.
So statement 2 adds nothing to statement 1, and it isn’t sufficient on its own either. The question asked about the hundreds digit of the sum x. Data about the units place of this addition problem is simply too many steps away because we can’t know what will happen in the tens place. Only statement 1 is sufficient on its own, and the correct answer is A.
Official place value problem
Here’s a final official place value problem for this article:
If n is a positive integer, what is the tens digit of n?
- The hundreds digit of 10n is 6.
- The tens digit of n + 1 is 7.
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.
I’ve seen smart math students get stumped by problems like this one. But the problem isn’t complicated; it’s just unfamiliar. Many test-takers are hesitant to model examples when the value isn’t known for sure. Where do you start? How do you know you aren’t “breaking the rules”? We must remember that the point of examples is not always to follow the rules but sometimes to find them.
This is one data sufficiency problem where statement 2 is easier to start with. If the tens digit of n + 1 is 7, can we determine the tens digit of n? Well, there are several scenarios in which the tens digit of n is also 7. If n + 1 is any integer between 71 and 79, inclusive, then the tens digit of n is 7. But if n + 1 is 70, n is 69 and has a tens digit of 6. Statement 2 is therefore insufficient for determining the tens digit of n.
Statement 1 tells us that the hundreds digit of 10n is 6. Let’s multiply some two-digit numbers by 10 and see what happens to the digits:
47 * 10 = 470
83 * 10 = 830
69 * 10 = 690
What’s happening should make sense: when an integer is multiplied by 10, we simply “tack on” a 0 to the right side of the number, and each digit is “pushed” one place to the left. The digit that was in the units place now occupies the tens place, the digit that was in the tens place now occupies the hundreds place, and so on and so forth. Since this is true, knowing the hundreds digit of 10n is just as good as knowing the tens digit of n. Statement 1 is also sufficient, and the correct answer is D.
This last problem belongs to a class of data sufficiency problems that ask about a certain digit of integer n (or some other variable) and then give statements about a digit occupying a different place of 10n or 1000n or n/100.
The same principle applies in all of them: multiplication or division by any power of 10 represents a “shifting” of the digits of the number. Multiplication moves the digits to the left by however many zeros are in the power of 10; division moves the digits to the right by however many zeros are in the power of 10. We know this, but the construction of the data sufficiency problems can make it hard to recognize that this is the property controlling the problem. Now you know!
In the next article, we’ll look at the interesting case of the digits in a two-digit integer switching places.
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Contributor: Elijah Mize (Apex GMAT Instructor)