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Remainders on the GMAT

Catching Sneaky Remainder Questions on the GMAT

BY David Goldstein, VERITAS PREP

One of my favorite topics to teach is remainders. We learn about remainders in grade school and when I introduce the topic in class, the response is often amused incredulity. It isn’t hard to see that when 16 is divided by 7, the remainder is 2. How can it possibly be the case that something we learned in fifth grade is included on a test that helps determine where we go to graduate school?

But in mathematics, seemingly basic topics often have broader applications. So let’s consider both simple and complex applications of remainders on the GMAT. The most straightforward scenario is for the question to ask what the remainder is in a given context. We’ll start by looking at an official Data Sufficiency question of moderate difficulty:

What is the remainder when x is divided by 3?
(1) The sum of the digits of x is 5
(2) When x is divided by 9, the remainder is 2

Pretty straightforward question. In Statement 1, we could approach by simply picking numbers. If the sum of the digits of x is 5, x could be 14. When 14 is divided by 3, the remainder is 2. Similarly, x could be 32. When 32 is divided by 3, the remainder will again be 2. Or x could be 50, and still, the remainder when x is divided by 3 will be 2. So no matter what number we pick, the remainder will always be 2. Statement 1 alone is sufficient.

Note that if we know the rule for divisibility by 3 – if the digits of a number sum to a multiple of 3, the number itself is a multiple of 3 – we can reason this out without picking numbers. If the sum of the digits of x were exactly 3, the remainder would be 0. If the sum of the digits of x were 4, then logically, the remainder would be 1. Consequently, if the sum of the digits of x were 5, the remainder would have to be 2.

Again, in Statement 2, we can pick numbers. We’re told that when x is divided by 9, the remainder is 2. To quickly generate a list of numbers that we might test, we can start with multiples of 9: 9, 18, 27, 36, etc. Then, we can add two to each of those multiples of 9 to get the following list of numbers: 11, 20, 29, 38, etc. All of these numbers will give us a remainder of 2 when divided by 9. Now we can test them. If x is 11, when x is divided by 3, the remainder will be 2. If x is 20, when x is divided by 3, the remainder will be 2. We’ll quickly see that the remainder will always be 2, so Statement 2 is also sufficient on its own. The answer to this question is D, either statement alone is sufficient. That’s not too bad. This question is discussed HERE.

But the GMAT won’t always be so conspicuous about what category of math it’s testing. Take this more challenging question, for example:

June 25, 1982 fell on a Friday. On which day of the week did June 25, 1987 fall. (Note: 1984 was a leap year.)
A) Sunday
B) Monday
C) Tuesday
D) Wednesday
E) Thursday

If you’re anything like my students, it’s not blindingly obvious that this is a remainder question in disguise. But that is precisely what we’re dealing with. Consider a very simple case. Say that June 1 is a Monday, and I want to know what day of the week it will be 14 days later. Clearly, that would also be a Monday. And if I asked you what day of the week it would be 16 days later, you’d know that it would be a Wednesday – two days after Monday. Put another way – because we’re dealing with weeks, or increments of 7 – all we need to do is divide the number of days elapsed by 7 and then find the remainder in order to determine the day of the week. 16 divided by 7 gives a remainder of 2, so if June 1 is a Monday, 16 days later must be 2 days after Monday.

Suddenly the aforementioned question is considerably more approachable. From June 25, 1982 to June 25, 1983 a total of 365 days will pass. 365/7 gives a remainder of 1, so if June 25, 1982 was a Friday, June 25 1983 will be a Saturday. From June 25, 1983 to June 25, 1984, 366 days will pass because 1984 is a leap year. 366/7 gives a remainder of 2, so if June 25, 1983 was a Saturday, June 25, 1984 will be 2 days later, or Monday. We already know that in a typical 365 day year, the remainder will be 1, so June 25, 1985 will be Tuesday, June 25, 1986 will be Wednesday and June 25, 1987 will be Thursday, which is our answer. This question is discussed HERE.

Takeaway: the challenge of the GMAT isn’t necessarily that questions are asking you to do difficult math, but that it can be hard to figure out what the questions are asking you to do. When you encounter something that seems unfamiliar or strange, remind yourself that virtually every problem you encounter will involve the application of a concept considerably simpler than the nebulous wording the question might suggest.
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Posts: 64242
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2 Tips to Make GMAT Remainder Questions Easy

BY David Goldstein, VERITAS PREP

At times, it will be helpful to know the kind of terminology we’re taught in grade school, while at other times, we’ll simply want to select simple numbers that satisfy the parameters of a Data Sufficiency statement.

So let’s explore each of these scenarios in a little more detail. A simple example can illustrate the terminology: if we divide 7 by 4, we’ll have 7/4 = 1 + ¾.

7, the term we’re dividing by something else, is called the dividend. 4, which is doing the dividing, is called the divisor. 1, the whole number component of the mixed fraction, is the quotient. And 3 is the remainder. This probably feels familiar.

In the abstract, the equation is: Dividend/Divisor = Quotient + Remainder/Divisor. If we multiply through by the Divisor, we get: Dividend = Quotient*Divisor + Remainder.

Simply knowing this terminology will be sufficient to answer the following official question:

When N is divided by T, the quotient is S and the remainder is V. Which of the following expressions is equal to N?
A) ST
B) S + V
C) ST + V
D) T(S+V)
E) T(S – V)

In this problem, N – which is getting divided by something else – is our dividend, T is the divisor, S is the quotient, and V is the remainder. Plugging the variables into our equation of Dividend = Quotient*Divisor + Remainder, we get N = ST + V… and we’re done! The answer is C.

(Note that if you forgot the equation, you could also pick simple numbers to solve this problem. Say N = 7 and T = 3. 7/3 = 2 + 1/3. The Quotient is 2, and the remainder is 1, so V = 1. Now, if we plug in 3 for T, 2 for S, and 1 for V, we’ll want an N of 7. Answer choice C will give us an N of 7, 2*3 + 1 = 7, so this is correct.) This question is discussed HERE.

When we need to generate a list of potential values to test in a data sufficiency question, often a statement will give us information about the dividend in terms of the divisor and the remainder.

Take the following example: when x is divided by 5, the remainder is 4. Here, the dividend is x, the divisor is 5, and the remainder is 4. We don’t know the quotient, so we’ll just call it q. In equation form, it will look like this: x = 5q + 4. Now we can generate values for x by picking values for q, bearing in mind that the quotient must be a non-negative integer.

If q = 0, x = 4. If q = 1, x = 9. If q=2, x = 14. Notice the pattern in our x values: x = 4 or 9 or 14… In essence, the first allowable value of x is the remainder. Afterwards, we’re simply adding the divisor, 5, over and over. This is a handy shortcut to use in complicated data sufficiency problems, such as the following:

If x and y are integers, what is the remainder when x^2 + y^2 is divided by 5?
(1) When x – y is divided by 5, the remainder is 1
(2) When x + y is divided by 5, the remainder is 2

In this problem, Statement 1 gives us potential values for x – y. If we begin with the remainder (1) and continually add the divisor (5), we know that x – y = 1 or 6 or 11, etc. If x – y = 1, we can say that x = 1 and y = 0. In this case, x^2 + y^2 = 1 + 0 = 1, and the remainder when 1 is divided by 5 is 1. If x – y = 6, then we can say that x = 7 and y = 1. Now x^2 + y^2 = 49 + 1 = 50, and the remainder when 50 is divided by 5 is 0. Because the remainder changes from one scenario to another, Statement 1 is not sufficient alone.

Statement 2 gives us potential values for x + y. If we begin with the remainder (2) and continually add the divisor (5), we know that x + y = 2 or 7 or 12, etc. If x + y = 2, we can say that x = 1 and y = 1. In this case, x^2 + y^2 = 1 + 1 = 2, and the remainder when 2 is divided by 5 is 2. If x + y = 7, then we can say that x = 7 and y = 0. Now x^2 + y^2 = 49 + 0 = 49, and the remainder when 49 is divided by 5 is 4. Because the remainder changes from one scenario to another, Statement 2 is also not sufficient alone.

Now test them together – simply select one scenario from Statement 1 and one scenario from Statement 2 and see what happens. Say x – y = 1 and x + y = 7. Adding these equations, we get 2x = 8, or x = 4. If x = 4, y = 3. Now x^2 + y^2 = 16 + 9 = 25, and the remainder when 25 is divided by 5 is 0.

We need to see if this will ever change, so try another scenario. Say x – y = 6 and x + y = 12. Adding the equations, we get 2x = 18, or x = 9. If x = 9, y = 3, and x^2 + y^2 = 81 + 9 = 90. The remainder when 90 is divided by 5 is, again, 0. No matter what we select, this will be the case – we know definitively that the remainder is 0. Together the statements are sufficient, so the answer is C. This question is discussed HERE.

Takeaway: You’re virtually guaranteed to see remainder questions on the GMAT, so you want to make sure you have this concept mastered. First, make sure you feel comfortable with the following equation: Dividend = Divisor*Quotient + Remainder. Second, if you need to select values, you can simply start with the remainder and then add the divisor over and over again. If you internalize these two ideas, remainder questions will become considerably less daunting.
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# Remainders on the GMAT  