The Remainder Cycle
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One common complaint I hear from my students is that they ‘haven’t done math like this since high school.’ And they’re pretty much right: the concepts in the Quant section are by and large wrapped up by Algebra II. But for some subjects, my students drastically underestimate how long it has been since they’ve thought about them. One such subject: remainders on the GMAT.
The last time you probably thought about remainders was early middle school. When you learned long division, you started with things that divided perfectly, and then you learned “25 ÷ 7 is 3 with 4 left over, so we say it is 3 remainder 4.” But then pretty quickly, you learned about fractions and decimals, so you just said “25 ÷ 7 is 3 and 4/7.” Eventually, you just started leaving it as ’25/7′ and didn’t even worry about it anymore. Remainders, like Destiny’s Child and the purest joys of childhood, faded into distant memory.
But now they’ve come screaming back into relevance in ways you’ve never thought about them before. The GMAT asks about remainders for a simple reason: they are patterned. Consider this: when dividing any integer by 7, what possible remainders are there? Test a few numbers and see. Could you ever get remainder 8?
You might realize that the only possible remainders are 1, 2, 3, 4, 5, and 6, but don’t forget you can also get remainder 0—for those numbers that are actually multiples of 7. Do you notice anything about the order in which those remainders come?
Remainders come in a cycle. When dividing by 7: 8 has remainder 1, 9 has remainder 2, 10 has remainder 3, 11 has remainder 4, 12 has remainder 5, 13 has remainder 6, 14 has remainder 0, 15 has remainder 1, 16 has remainder 2, 17 has remainder 3… etc. etc. etc., onward and forever. That is, when dividing by 7, the remainders cycle from 0-6 continuously. This is a general rule:
When dividing integers by integer n, there are n possible remainders, 0 through (n-1), and these remainders cycle in an infinite loop.
I call this the ‘remainder cycle.’ Take a look at the table below to see it in practice for a few smaller numbers. Note that you can say something like “2 ÷ 9 is 0 remainder 2,” so it actually is said to have a remainder 2.
Remainder Cycles When Dividing by Integers
Do you see that when you are dividing by n, the same remainder appears in spacings of n? For example, 5, 9, 13, 17, 21, 25 all have remainder 1 when dividing by 4, and they’re all 4 apart. This is another rule worth tucking away:
If an integer x has a remainder r when divided by n, then x +/- any multiple of n will also have remainder r.
It’s good to be familiar with this general concept of the remainder cycle, but there are two particular varieties the GMAT really gravitates towards. One of them you’re likely very aware of, though you probably have never thought about it in this way. Here’s a hint: it has to do with the remainder cycle when dividing by 2.
You can see that the only possible remainders when dividing by 2 are 0 and 1. We have names for these kinds of numbers: we call them ‘even’ or ‘odd.’ Odd numbers are integers that have remainder 1 when divided by 2, and even numbers have remainder 0 when divided by 2 (that is, they are multiples of 2). You know—or you will soon, as you study for the GMAT—that even and odd numbers are subject to certain rules and patterns when you add, subtract, multiply, or divide them (for example: even*odd = even, odd*odd = odd, odd +/- even = odd). The GMAT has many questions dealing with ‘even and oddness,’ though very rarely in the terms of remainders—at least explicitly. But it turns out the remainder cycle is behind all the patterns in odds and evens.
However, the GMAT also has a secret affinity for the remainder cycle when dividing by 3. Now there are no labels like ‘even’ and ‘odd’ for these numbers, so I like to use ‘R0,’ ‘R1,’ and ‘R2’ for numbers that have remainder 0, 1, and 2 when dividing by 3 (R0s will be multiples of 3). Just like even and odd are the only ‘types’ of numbers when dividing by 2, these are the only ‘types’ of numbers when dividing by 3, and there are rules and patterns for these numbers as well. For instance, R1 + R2 = R0; R1 +/- R0 = R1; R2 * R2 = R1, R2 + R2 = R1, R0 * (any integer) = R0. There are several more, but you don’t need to find and memorize them; just know that they exist, and any time a problem asks about divisibility by 3, realize you’ll probably need to consider these three types of numbers.
Here’s a practice problem.
If n is an integer, which of the following must be divisible by 3?
A) n^3 – 4n
B) n^3 + 4n
C) n^2 +1
D) n^2 -1
E) n^2 -4
Since there are variables in the answer choices, you might realize you can plug in numbers. But how do you decide which? Well, since we’re asked about divisibility by 3, we should probably plug in an R0, R1, and R2 (just like how on problems about even/odd, it’s best to plug in an even and an odd).
If you plug in one of each type, the only answer choice that is always divisible by 3 is A.
And here’s one final bonus fact about remainders:
If integer x has remainder r when divided by n, then the remainder when px is divided by n is equal to the remainder when pr is divided by n.
For example: 13 has remainder 1 when divided by 6. What is the remainder when 13 * 31 is divided by 6? Well, the remainder of 13 * 31 will be the same as the remainder of 1 * 31 when dividing by 6. And the remainder of 31 when divided by 6 is 1.
So welcome back to middle school! Now that remainders have returned to your life, social ostracism, acne, and raging hormones are sure to follow. ?
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Reed Arnold
is a Manhattan Prep instructor based in New York, NY. He has a B.A. in economics, philosophy, and mathematics and an M.S. in commerce, both from the University of Virginia. He enjoys writing, acting, Chipotle burritos, and teaching the GMAT. Check out Reed’s upcoming GMAT courses here.