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Study Guide

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In life, a picture is worth a thousand texts, a good shortcut is worth extra time on Minecraft, and an outline can clarify any project. In math, we tend to spend lots of time reducing, simplifying, and dividing, too.

Divisibility shortcuts reduce, simplify, and cut work in half like you wouldn't believe. They help us simplify fractions faster, divide faster, and ace math classes faster, thus getting us back to playing video games, building roller coasters, and trying to take over the world.

Divisibility rules are clues to help us know if we can divide two numbers evenly or not. There are divisibility rules for most numbers, but we'll just look at the numbers 2 through 12 right now.

Let's start with 2.

We know a number is divisible by 2 if it can sing a duet. Just kidding. No number can sing a duet alone. A number is divisible by 2 if its last digit is even. For example, 8384 is divisible by 2 because its last digit, 4, is an even number. On the other hand, 4231 is *not* divisible by 2 because its last digit, 1, is as odd as a cat wearing a leash.

To find out if a number is divisible by 3, we add. A number is divisible by 3 if the sum of its digits is divisible by 3. Let's use 816 as an example. Since the sum of its digits is 8 + 1 + 6 = 15, and 15 is divisible by 3, then the entire number 816 is divisible by 3. But what about 263? Since 2 + 6 + 3 = 11, and 11 is definitely not divisible by 3, then 263 isn't divisible by 3 either.

The divisibility test for 4 is based on the motto, "It's not how you start but how you finish that matters." A number is divisible by 4 if its last two digits are divisible by 4. Let's use 89,016 as an example. Since its last two digits are 16, and 16 is divisible by 4, the entire number 89,016 is divisible by 4. The number 3533 isn't as lucky. The last two digits in 3533 are 33, but 33 isn't divisible by 4, so we know the entire number fails the divisible-by-4 test. Bummer.

Deciding whether a number is divisible by 5 is pretty simple stuff. If a number ends in 0 or 5, it's divisible by 5. If it ends in any other number, it isn't divisible by 5. Boom.

The divisibility test for 6 relies on the divisibility tests for 2 and 3. If a number is divisible by *both* 2 *and* 3, then it's divisible by 6 too. Remember, 2 *and* 3 or nothing. For example, is the number 216 divisible by 6? Welp, it ends with an even number (6), so we know it's divisible by 2. When we add the digits, we get 2 + 1 + 6 = 9, which is divisible by 3, so we also know that 216 is divisible by 3. Since we just showed that 216 is divisible by both 2 and 3, it must also be divisible by 6.

The divisibility test for 7 is a bit more complicated. To figure out whether a number is divisible by 7, we double the last digit, then subtract it from the rest of the number. If the result is divisible by 7, then the whole number is divisible by 7. Take the number 271, for instance. The last digit (1) doubled is 2. When we subtract 2 from 27, we get 25. Since 25 isn't divisible by 7, then the number 271 isn't divisible by 7 either. Let's look at the number 224. If we double the last digit (4), we get 8. Next, we subtract 22 – 8, which is 14. Since 14 is divisible by 7, then so is 224.

The divisibility test for 8 is similar to the divisibility test for 4, plus a digit. If the last *three* digits of a number are divisible by 8, then the entire number is divisible by 8. Let's take 23,888. Since 888 is divisible by 8, the entire number is divisible by 8.

We're on a roll here. How 'bout 9? The divisibility test for 9 is just like the test for 3. If the sum of its digits is divisible by 9, then the entire number is divisible by 9. Let's use 981 as an example. Since 9 + 8 + 1 = 18, and 18 is divisible by 9, then the entire number 981 is divisible by 9 as well.

The divisibility test for 10 is yet another example of why zero is our hero. If a number ends in 0, it's divisible by 10. If it ends in anything other than 0, it's a non-hero, not-divisible-by-10 number. Harsh.

The divisibility test for 11 can get a little wild. To decide whether a number is divisible by 11, we subtract, then add, each digit of the number from left to right until we have a final value. If that final value is divisible by 11, then the entire number is divisible by 11. Is 2291 divisible by 11? First, we find the alternate sum of digits like this (starting with subtraction): 2 – 2 + 9 – 1 = 8, Since 8 is not divisible by 11, then 2291 is not divisible by 11 either. Let's try it out with 10,802. Since 1 – 0 + 8 – 0 + 2 = 11, and 11 is divisible by 11, we know that 10,802 is divisible by 11. Pretty cool, huh?

The divisibility test for 12 relies on our good friends 3 and 4. If a number is divisible by both 3 and 4, it's divisible by 12 too. Let's say we want to know if 8724 is divisible by 12. We check if it's divisible by 3, then we check if it's divisible by 4 (or the other way around). If it passes both checks, then it's divisible by 12. Adding the digits gives us 8 + 7 + 2 + 4 = 21 and 21 is divisible by 3, so 8724 is divisible by 3. Now, we've gotta check to see if it's divisible by 4. Since the last two digits of 8724 are 24 and 24 is divisible by 4, we know that 8724 is divisible by 4. Since 8724 passes the divisibility tests for both 3 and 4, we know it's divisible by 12.

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