Remainders: Small Numbers Reveal the Pattern

Say you see this on Test Day: When the positive integer n is divided by 5, the remainder is 3. What is the remainder when 4n is divided by 5? I watch people do the same thing with a question like this every week. They freeze. They try to remember some rule from a number theory class they never took. They write n = 5q + 3, stare at it, and push symbols around. Two minutes later, they guess.

Here is my blunt verdict: remainder questions are NOT hard. They LOOK hard because they sound like math you were never taught. But the GMAT is not checking whether you know number theory. It is checking whether you will stop, pick a few small numbers and look. That's it.

Watch. A number that leaves 3 when you divide it by 5 could just be 3 (zero full groups of 5, with 3 left over). So n = 3. Then 4n = 12. Divide 12 by 5 and you get 2 full groups with 2 left over. Try a second number to be safe: n = 8. Then 4n = 32, which is 6 groups of 5 (that's 30) with 2 left over. Same answer twice. The remainder is 2. Total time: maybe 20 seconds, and you never wrote a single variable.

First, what a remainder actually is

A remainder is what's left over after you divide out as many full groups as you can. Divide 17 by 5: you get 3 full groups of 5 (that's 15), and 2 are left over. The remainder is 2. The number you divide by (here, 5) is called the divisor.

Two rules you need, and only two:

• The remainder is ALWAYS smaller than the divisor. Divide by 5 and the only possible remainders are 0, 1, 2, 3 and 4.
• The remainder can be 0. That just means the number divides evenly.

Why small numbers work

The question hands you a rule, like "remainder 3 when divided by 5." EVERY number that follows that rule gives the same answer to the question. So why would you pick a hard one? Pick the easiest numbers that fit. It's the same move as picking 100 on a percentage question: you swap a scary abstract problem for simple arithmetic.

The numbers that leave 3 when divided by 5 are 3, 8, 13, 18, 23 and so on. Stop reading and think about this for a second. Why does that list jump by exactly 5 each time? Don't read another word until you have a reason.

Here it is: adding one more full group of 5 changes nothing about what's left over. That's why the list always climbs by the divisor. And that's why any number on the list works just as well as any other.

On the GMAT Focus Edition, you get 21 Quant questions in 45 minutes, all of them Problem Solving (regular multiple choice with five answers), according to GMAC's page on the exam format. That's an average of a little over 2 minutes per question. Listing four small numbers takes 30 seconds. Writing equations you half remember takes a lot longer, and you end up guessing anyway.

Remainders hiding inside giant powers

Sometimes the number is so big you can't possibly work it out. Good. That's the GMAT telling you there's a pattern. Small powers will show it to you.

Remainders in Data Sufficiency

Remainders also show up in the Data Insights section, inside Data Sufficiency questions. In Data Sufficiency you don't solve the question. You decide whether the two facts you're given (called statements) give you ENOUGH information to answer it. Testing values is even more powerful here, because one pair of numbers that give different answers proves a statement isn't enough.

Notice what both examples had in common. No theorems. No clever tricks. A short list on paper, and the pattern showed up on its own. When a question asks whether something divides evenly, it's really a question about prime factors, and the same habit applies: write it out.

The mistakes I watch people make every week

1. Writing an equation and staring at it. "n = 5q + 3" is true. It's also useless to most people under a clock. INSTEAD: List the first four numbers that fit the rule and use them.
2. Skipping the smallest number. "Remainder 3 when divided by 5" and people start at 13 or 18. Bigger numbers mean more arithmetic and more chances for a silly mistake. INSTEAD: Start with the remainder itself. It's always the smallest number on the list.
3. Testing ONE number and calling it done. One number can get lucky. INSTEAD: Test at least two on Problem Solving, and three or more on Data Sufficiency.
4. Calling the pattern before it repeats. Two results don't make a cycle. INSTEAD: Keep writing until you see the pattern start over at least once.
5. Doing it in your head. This is pride, and it's expensive. You can review and change up to 3 answers per section on this test, but don't plan to spend those fixing mental math you never should have done. INSTEAD: Every number goes on the pad. EVERY one.

I'll bet that the next time you see a remainder question, you'll have the pattern on paper before the person next to you finishes writing their first equation. Not because you're better at math. Because you stopped trying to be a mathematician and started acting like a strategist.

Your move

Today, pull five remainder questions from your official practice materials. Before you look at any answer choices, write out four small numbers that fit each rule. Time yourself on every one. Then go back and redo the ones you missed using ONLY the list, no equations allowed. If any question took you more than 2 minutes, find the exact line on your pad where you stopped listing and started guessing. That line is the thing to train.