Here are 10 important divisibility rules you should know:
1 may be the loneliest number but it’s also a very important number for divisibility! Every integer is divisible by 1, and the result of any integer x divided by 1 is just x (when you divide an integer by 1, it stays the same). For example, a question might ask (as at least one official problem does): Does integer x have any factors y such that 1 < y < x?
Because every integer is divisible by itself and 1, that question is really just asking, “Is x prime or not?” because if there is a factor y that’s between 1 and x, x is not prime, and there is not such a factor, then x is prime. That “> 1” caveat in the problem may seem obtuse, but when you understand divisibility by 1, you can see that the abstract question stem is really just asking you about prime vs. not-prime as a number property.
A number is divisible by 2 if that number is even (and a number is even if it’s divisible by 2). That means that if an integer ends in 0, 2, 4, 6, or 8, you know that it’s divisible by 2. And here’s a somewhat-surprising fact: the number 0 is even! 0 is divisible by 2 with no remainder (0/2 = 0), so although 0 is neither positive nor negative it fits the definition of even and should therefore be something you keep in mind because 0 is such a unique number.
Because any even number is divisible by 2 (which also means that it can be written as 2 times an integer), an even number multiplied by any integer will keep 2 as a factor and remain even. So even x even = even and even x odd = even.
It’s been said that good things come in 3s, and divisibility rules are no exception! The divisibility rule for 3 works much like a magic trick and is one that you should make sure is top of mind on test day to save you time and help you unravel tricky numbers. The rule: if you sum the digits of an integer and that sum is divisible by 3, then that integer is divisible by 3. For example, consider the integer 219. 2 + 1 + 9 = 12 which is divisible by 3, so you know that 219 is divisible by 3 (it’s 3 x 73).
This rule can help you in many ways. If you were asked to determine whether a number is prime, for example, and you can see that the sum of the digits is a multiple of 3, you know immediately that it’s not prime without having to do the long division to prove it. Or if you had a messy fraction to reduce and noticed that both the numerator and denominator are divisible by 3, you can use that rule to begin reducing the fraction quickly.
Presidential Election and Summer Olympics enthusiasts, be four-warned! You already know the divisibility rule for 4: take the last two digits of an integer and treat them as a two-digit number, and if that’s divisible by 4 so is the whole number. So for 2016 – next year and that of the next presidential election and Brazil Olympics – the last two-digit number, 16, is divisible by 4, so you know that 2016 is also divisible by 4.
If you fail to see immediately that a number is divisible by 4 given that rule, fear not! Being divisible by 4 just means that a number is divisible by 2 twice. So if you didn’t immediately see that you could factor a 4 out of 2016 (it’s 504 x 4), you could divide by 2 (2 x 1008) and then divide by 2 again (2 x 2 x 504) and end up in the same place without too much more work.
Who needs only 5 fingers to divide by 5? All of us – divisibility by 5 is so easy you should be able to do it with one hand tied behind your back! If an integer ends in 5 or 0 you know that it’s divisible by 5 (and we’ll talk more about what extra fact 0 tells you in just a bit…).
Your favorite character from the hit 1990’s NBC sitcom “Blossom” is also an easy-to-use divisibility rule! Since 6 is just the product of 2 and 3 (2 x 3 = 6), if a number meets the divisibility rules for both 2 (it’s even) and 3 (the sum of the digits is divisible by 3) it’s divisible by 6. So if you need to reduce a number like 324, you might want to start by dividing by 6, instead of by 2 or 3, so that you can factor it in fewer steps.
Ah, magnificent 7. While there is a “trick” for divisibility by 7, 7 occurs much less frequently in divisibility-based problems (as do other primes like 11, 13, 17, etc.), so 7 is a good place to begin to think about a strategy that works for all numbers, rather than memorizing limited-use tricks for each number. To test whether a large number, such as 231, is divisible by 7, find an obvious multiple of 7 nearby and then add or subtract multiples of 7 to see whether doing so will land on that number. For 231, you should recognize that a nearby multiple of 7 is 210 (you know 21 is 7 x 3, so putting a 0 on the end of it just means that 210 is 7 x 30). Then as you add 7s to get there, you go to 217, then to 224, then to 231. So in your head you can see that 231 is 3 more 7s than 7 x 30 (which you know is 210), so 231 = 7 x 33.
8 is enough! As you saw above with 4s and 6s, when you start working with non-prime factors it’s often easier to just divide out the smaller prime factors one at a time than to try to determine divisibility by a larger composite number in one fell swoop. Since 8 = 2 x 2 x 2, you’ll likely find more success testing for divisibility by 8 by just dividing by 2, then dividing by 2 again, then dividing by 2 a third time. So for a number like 312, rather than working through long division to divide by 8, just divide it in half (156) then in half again (78) then in half again (39), and you’ll know that 312 = 39 x 8.
While “nein” may be German for “no,” you should be saying “yes” to divisibility by nine! 9 shares a big similarity with 3 in that a sum-of-the-digits rule applies here too. If you sum the digits of an integer and that sum is a multiple of 9, the integer is also divisible by 9. So, for example, with the number 729, because 7 + 2 + 9 = 18, you know that 729 is divisible by 9 (it’s 81 x 9, which actually is 9 to the 3rd power).
We’ve saved the best for last! If a number ends in 0, it’s divisible by 10, giving you a great opportunity to make the math easy. For example, a number like 210 (which you saw above) lets you pull the 0 aside and say that it’s 21 x 10, which means that it’s 3 x 7 x 10.
Working with 10s makes mental (or pencil-and-paper) math quick and convenient, so you should seek out opportunities to use such numbers in your calculations. For example, look at 693: If you add 7, you get to a number that ends in two 0s (so it’s 7 x 10 x 10), meaning that you know that 693 is divisible by 7 (it’s 7 away from an easy multiple of 7) *and* that it’s 7 x 99 because it’s one less 7 than 7 x 100. It’s a good idea to quickly check for, “If I have to add x to get to the nearest 0, then does that give me a multiple of x?” (297 is 3 away from 300, so you know that 297 = 99 x 3). And since 10 = 2 x 5, it’s also helpful sometimes to double a number that ends in 5 (try 215, which times 2 = 430) to see how many 10s you have (43). That tells you that 215 = 43 x 5 because 215 x 2 = 43 x (2 x 5). Working with 10s can make mental math extremely quick – we’d rate numbers that end in 0 a perfect 10!