Remainders

The SBC GMAT Files

Loneliest Numbers:  Pizza, Dance Partners, And Remainders On The GMAT
Jeff Rogers, GMAT Specialist, Test Prep New York – San Francisco

Most GMAT test takers haven’t thought much about remainders since their days of scratching out long division in grade school workbooks.  Test makers, on the other hand, have done a great deal of thinking about remainders as an important aspect of the larger subjects of divisibility and number properties, and the GMAT Quant section reflects that fact.

Although most of us are disinclined to pay it much mind in day-to-day life, the basic idea of a remainder pops up constantly.  Consider the cold arithmetic of dividing an eight-slice pizza between six friends:  everybody gets one slice with two left over.  That’s a remainder.  Consider the even colder arithmetic of a dance class ”¨(or a dance, even) with 17 people all looking for a practice partner.  That means eight pairs of partners with one person left out.

In the simplest of terms, then, a remainder is what’s left over when one number is divided by another number that doesn’t divide the first number evenly. Using slightly less simple terms, we can say that when a number evenly divides another number, we have a quotient with no remainder (e.g.,  = 2, where 2 is the quotient and the remainder is 0).  If we were to divide that 8 by 3, however, we still have a quotient of 2 (because there are 2 multiples of 3 that are less than 8), but now we also have a remainder of 2.  This can be expressed as = 2 r 2.  The remainder, here, is 2 because 8 is two more than the largest multiple of 3 that divides 8.

This is where things begin to get more complicated.  We can express the answer to the question  = ? as 2 r 2 (as demonstrated), but we can also say that = 2 and or even that = 2.667.  In the former case, we’re saying that the number 8 contains two 3’s and two-thirds of an additional 3.  The latter case simply expresses the same idea as a decimal rather than a fraction.  In other words, 8 divided by 3 is the same as the improper fraction , which is equivalent to the mixed number 2 and , and to the decimal 2.667.  When we convert a decimal, say 2.25, into an improper fraction, we are using the idea that 2.25 is equal to 2 and , which is equal to . The takeaway here is that any number divided by another number to yield a quotient and a remainder can be expressed with a decimal or a fraction.

The GMAT loves to play with these ideas and equivalences, and requires test takers to think abstractly about remainders.  A GMAT favorite is to ask a question along these lines:

If there is a remainder of 2 when x is divided by 5 and x divided by 7 yields a remainder of 3, which of the following could be x?

A)    11
B)    17
C)    42
D)    94
E)    113

In plainer English, the GMAT is telling us that x is two larger than a multiple of 5 and three larger than a multiple of 7.  The easiest way to find x, then is to make a list of numbers that are two greater than a multiple of five and another list of numbers that are three greater than a multiple of 7 and see where the lists overlap.

You know the multiples of 5:  5, 10, 15, 20, 25, 30, 35, 40, and so on””every number that ends in a 5 or a 0.  The numbers that are two greater than a multiple of 5 will be all of those multiples plus 2, i.e., 7, 12, 17, 22, 27, 32, 37, 42, and so on.  Thinking critically, we might notice here that all of the numbers that are two greater than a multiple of five (or yield a remainder of 2 when divided by 5) end in a 2 or a 7.  This would be a great way to eliminate three answer choices quickly””(A), (D), and (E).

The process is the same for finding the numbers that yield a remainder of 3 when divided by 7.  Begin with the multiples of 7 and add three to each.  We start with 7, 14, 21, 28, 35, 42, 49, and so on, and adding three, we come up with 10, 17, 24, 31, 38, 45, 52.  Cross-checking this list with our first one, we see that the only possible value for x among the answer choices is (B) 17.

If you can do another problem like this one, you can crack the remainder questions on the GMAT.  Some will be a little more complicated, but they’ll all build on the same set of ideas.  Get them down cold, and next time you’re carving up an eight-slice pizza with five friends tell them that the two extra slices goes to the first person who can name the smallest number that is six less than a multiple of 7 and yields a remainder of 2 when divided by 5.

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