GMAT Divisibility From Scratch
Divisibility questions are usually testing whether you can eliminate answer choices fast. The shortcuts for 2, 3, 4, 5, 8, 9 and 10 are worth knowing cold. The more valuable skill is having a method for the numbers with no shortcut at all, like 13 and 17.
The same idea, four names
A GMAT question can express one relationship four different ways, and it will switch between them:
- 10 is divisible by 2
- 2 is a factor of 10
- 2 divides 10
- 10 is a multiple of 2
All four mean 10 = 2 × 5, with the second factor being a whole number. Recognizing them as the same thing saves you re-reading the question.
The shortcut table
| Divisor | Check |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Digits sum to a multiple of 3 |
| 4 | Last two digits form a multiple of 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Passes both the 2 test and the 3 test |
| 8 | Last three digits form a multiple of 8 |
| 9 | Digits sum to a multiple of 9 |
| 10 | Last digit is 0 |
Two things people mix up. The 5 and 10 rules are different: 5 allows 0 or 5, 10 requires 0. And the digit-count rules escalate: one digit for 2 and 5, two for 4, three for 8.
Why the digit rules work: anything beyond the last two digits ends in 00, which is divisible by 100 and therefore by 4. Same logic for 8 and 1,000.
With the 9 test you can repeat the sum. For 999: 9 + 9 + 9 = 27, then 2 + 7 = 9. Divisible.
Add or subtract two multiples and you get another
If two numbers are both divisible by n, then their sum and their difference are too. 12 and 18 are both divisible by 6, and so are 30 and 6.
This looks trivial with small numbers, and it is the foundation of everything useful that follows.
The split method: for 13, 17, and anything else
This is the most valuable technique on the page, and the one that separates a fast answer from a stuck one.
When no shortcut exists, split the number into two parts you can each check separately.
Is 1,326 divisible by 13? Split it as 1,300 + 26. 1,300 is 13 × 100. 26 is 13 × 2. Both parts divide, so the whole does.
Choose the split deliberately: pick a nearby number that is obviously a multiple of your divisor, and check what is left over. You can split more than twice if it helps. 510 can go to 400 + 80 + 30 when testing for 8.
It disproves as well. 1,328 splits as 1,300 + 28. The first part divides by 13, the second does not, so the whole does not. For this to work, one part must divide and the other must not. Two parts that both fail tell you nothing.
Chaining
If a divides b and b divides c, then a divides c.
Useful in two situations. With unknowns: if 5 divides x and x divides y, then 5 divides y. And as a shortcut: once you have shown a number is divisible by 8, you do not need to test 4, since 4 divides 8.
Order your checks to eliminate fastest
On a question asking which number is divisible by 4, 5, 8, 9 and 13, you are not verifying all five for all five options. You are eliminating.
Start with the cheapest test that kills the most answers. The 5 test is one glance at the last digit. Then 8, which is harder but eliminates several. Skip 4 entirely once 8 is confirmed. Leave 13 until last, because the split method costs the most time and by then only two candidates remain.
Choosing the order is the actual skill being tested.