It almost never fails when I’m working with a new student that the first time a question involving remainders rears its ugly head, we need to spend some time talking about what remainders are and how to find them. This happens fairly early in the process with me, intentionally. I always start with new students by working through the plug in and backsolve chapters in my math book, and the plug in chapter prominently features a remainder question. That’s what’s discussed in the video above.
A remainder, just so we’re clear, is what’s left over when one positive integer is divided by another. When 17 is divided by 5, for example, the remainder is 2: 5 goes into 12 three times—making 15—but there’s still 2 left over to get to 17. Remember long division?
Remainders are always whole numbers, never decimals. However, there’s a handy shortcut that’ll allow you to convert decimals to remainders:
- Do the division on your calculator.
- If there’s no decimal, then there’s a remainder of 0. If there is a decimal, then there’s a remainder.
- Convert the decimal to a remainder by subtracting the part before the decimal point from the quotient you have on your calculator.
- You’ll be left with only a decimal. Multiply that decimal by the original divisor.
- Boom! You’ve got a remainder.
Here, I’ll show you. Let’s do 17/5 again. When you put 17/5 into your calculator, you get 3.4. Subtract 3 from that, and you’re left with 0.4. Multiply that by 5, and you’re left with 2—that’s your remainder! Note that if there’s a repeating decimal, you shouldn’t round it or you won’t get an integer remainder.
Here’s another example with the exact keystrokes I enter into my TI-83, and what the calculator displays.
What is the remainder when 52,343 is divided by 92?
52343/92 [ENTER] 568.9456522 Ans–568 [ENTER] .9456521739 Ans*92 [ENTER] 87
Therefore, 52,343/92 = 568 R 87.
Cool right? Any remainder operations you’ll be doing on the SAT will be far less involved and easily done with long division, so you don’t need to memorize this trick, but it’s there for you if you want it.
Is that all?
If you end up having to deal with remainders on your SAT, you’ll almost definitely have to do more than just divide two integers and find the remainder. You’ll probably be asked (as you are in the problem featured in the video above) to figure out something about unknown constants given some information about remainders. When that happens to you, here are the things it’s important to know:
If n is divided by k and leaves a remainder of r, then n must be r greater than a multiple of k. For example, if a number divided by 8 leaves a remainder of 3, then that number must be 3 greater than a multiple of 8. You’ll do well to plug in a nice, low number that fits that description, like 8 + 3 = 11.
The greatest possible remainder when dividing by k is k – 1. For example, if you’re dividing by 10, then the greatest possible remainder you can get is 9.
When you divide a bunch of consecutive integers by the same divisor k, the remainders will form a repeating pattern of consecutive integers from 0 to k – 1. For example, when you divide a bunch of consecutive integers by 3, you’ll get a repeating pattern like: 0, 1, 2, 0, 1, 2, 0, 1, 2, … The pattern might begin with any of those numbers, depending on which consecutive integer you begin dividing by 3, but the pattern will be the same.
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