Working With Remainders

March 12, 2010

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While they may sound simple, remainders can be a tricky concept. The GMAT requires you to understand how they work at a much more abstract level than your teacher did in fourth grade when you first learned what they were. Let me show you how best to think about them.

First, some terminology. When x is divided by y, that's the same as a fraction: "x over y."

  • The "quotient" is the whole number result: when 6 is divided by 3, the quotient is 2. When 8 is divided by 3, the quotient is also 2.
  • The remainder is what's left over. When 6 is divided by 3, the remainder is 0, because 3 is a factor of 6. When 8 is divided by 3, the remainder is 2, because there are 2 "left over" after you divide 3 into 8 two times.

It's perhaps easiest to see what a remainder is in terms of a mixed number. 7 divided by 3 is the same as the fraction 7/3, which is the same as the mixed number 2 1/3. The quotient is the whole number (2), while the remainder is the numerator of the fraction (1). The denominator of the fraction is the number we originally divided by.

Working With Variables

Now let's try it with unknowns. Let's say that a divided by b has a quotient of c and a remainder of d. That translates into the following equation:

a/b = c + d/b

If this is a new concept to you, it's worth memorizing. It doesn't come up frequently, but it's the sort of thing that would take plenty of time to puzzle out if you were confronted by it on test day.


Any number with a remainder could be expressed as a decimal, as well. For instance, 5 divided by 4 is 1 1/4, or 1.25. It's fairly straightforward to get from a remainder (or a fraction) to a decimal, but much less obvious how to work in the other direction.

In fact, if you're given a number in decimal form, you can't determine the exact remainder. However, you can come up with the set up possible remainders. For example, consider the number 5.24. That's the same as 5 + 0.24, or 5 + 24/100.

Is the remainder 24? It could be: if the original problem were 524 divided by 100, that would be correct. However, there are other possibilities. You can discover these by manipulating the fractional part, 24/100.

You can simplify 24/100 down to 6/25. Since 5 + 24/100 = 5 + 6/25, the remainder could be 6. If the original problem were 131 divided by 25, the result would be 5 with a remainder of 6. Since we can't simplify the fraction any further, 6 is the smallest possible remainder.

There are an infinite number of possible remainders for this example, but they can be easily classified. Once you find the smallest possible remainder (in this case, 6), you know that the remainder must be a multiple of that number. So, in the example, the remainder could be 6, 12, 18, 24, 30, etc.


To see a few test-like remainder questions, see Data Sufficiency #148 in the Official Guide (Orange Book), Problem Solving #60 in the Quantitative Review (Green Book), or Problem Solving #13 in the Diagnostic Test of the Official Guide. There are also a couple of remainder questions in my Algebra: Challenge problem set.



About the author: Jeff Sackmann has written many GMAT preparation books, including the popular Total GMAT Math, Total GMAT Verbal, and GMAT 111. He has also created explanations for problems in The Official Guide, as well as 1,800 practice GMAT math questions.

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