Divisibility: Find out if 3 divides evenly into an integer
Quick: What English word has 12 letters, almost half of which are are the letter “i” — well, 5 of the 12, to be exact?
Why it’s the word “D-I-V-I-S-I-B-I-L-I-T-Y” — a great thing to understand if you’re going to spend any amount of time doing math. And guess what: virtually ALL students do a fair amount of math, so everyone would do well to master the tricks of divisibility.
With the tricks for divisibility in your command, you will have a much easier time:
— reducing fractions
— multiplying fractions
— dividing fractions
— adding and subtracting fractions
— finding the GCF and LCM
— simplifying ratios
— solving proportions
— factoring algebraic expressions
— factoring quadratic trinomials
— Need I say more?
I’m sure you get the point — divisibility tricks are handy to know.
Since the tricks of divisibility are fun and interesting, too, I’ll share as many as I can think of. If, after I’m done, you know tricks I have not mentioned, feel free to share them as comments. Or, if you know any additional tricks for the numbers I’m covering, share those! It’s always fun to learn ways to get faster at math.
Today, I’ll share the trick that tells us whether or not a number is divisible by 3. Now many of you probably know the basic trick. But even if you do, don’t skip this blogpost. For after I show how this trick is usually presented, I’ll share a few extra tricks that most people don’t know, tricks that make the basic trick even easier to use.
Here’s how the trick is usually presented.
Take any whole number and add up its digits. If the digits add up to a multiple of 3 (3, 6, 9, 12, etc.), then 3 divides into the original number. And if the digits add up to a number that is not a multiple of 3 (5, 7, 8, 10, 11, etc.), then 3 does not divide into the original number.
Example A: Consider 311.
Add the digits: 3 + 1 + 1 = 5
Since 5 is NOT a multiple of 3, 3 does NOT divide into 311 evenly.
Example B: Consider 411.
Add the digits: 4 + 1 + 1 = 6
Since 6 IS a multiple of 3, 3 DOES divide into 411 evenly.
Check for yourself:
311 ÷ 3 = 103.666 … So 3 does NOT divide in evenly.
But 411 ÷ 3 = 137 exactly. So 3 DOES divide in evenly.
Isn’t it great how reliable math rules are? I mean, they ALWAYS work, if the rule is correct. In what other field do we get that level of certainty?!
Now, to make the rule work even faster, consider this trick. If the number in question has any 0s, 3s, 6s, or 9s, you can disregard those digits. For example, let’s say you need to know if 6,203 is divisible by 3. When adding up the digits, you DON’T need to add the 6, 0 or 3. All you need to do is look at the 2. Since 2 is NOT a multiple of 3, 3 does NOT go into 6,203.
So now try this … what digits do you need to add up in the following numbers? And, based on that, is the number divisible by 3, or not?
a) 5,391: Consider only the 5 & the 1. DIVISIBLE by 3.
b) 16,037: Consider only the 1 & 7. NOT divisible by 3.
c) 972,132: Consider only the 7, 2, 1 & 2. DIVISIBLE by 3.
Just as you can disregard any digits that are 0, 3, 6, and 9, we can also disregard pairs of numbers that add up to a sum that’s divisible by 3. For example, if a number has a 5 and a 4, we can disregard those two digits, since they add up to 9. And if a number has an 8 and a 4, we can disregard them, since they add up to 12, a multiple of 3.
Try this. See which digits you need to consider for these numbers. Then tell whether or not the number is divisible by 3.
a) 51,954: Disregard 5 & 1 (since they add up to 6); disregard the 9; disregard the 5 &4 (since they add up to 9). So number is DIVISIBLE by 3. [NOTE: If you can disregard all digits, then the number IS divisible by 3.]
b) 62,497: Disregard 6; disregard 2 & 4 (Why?); disregard 9. Consider only the 7. Number is NOT divisible by 3.
c) 102,386: Disregard 0, 3, 6. Disregard 1 & 2 (Why?). Consider only the 8. Number is NOT divisible by 3.
See how you can save time using these corollaries?
Using the trick and the corollaries, determine which numbers you need to consider, then decide whether or not 3 divides into these numbers.
a) 47: Consider the 4 and 7. Number NOT divisible by 3.
b) 915: Consider no digits. Number IS divisible by 3.
c) 4,316: Consider the 4, 1. Number NOT divisible by 3.
d) 84,563: Consider only the 5. Number NOT divisible by 3.
e) 71,031: Consider the 7, 1, 1. Number IS divisible by 3.
f) 367,492: Consider only the 7. Number NOT divisible by 3.
g) 5,648: Consider only the 5. Number NOT divisible by 3.
h) 12,039: Consider no digits. Number IS divisible by 3.
i) 79: Consider only the 7. Number NOT divisible by 3.
j) 617: Consider the 1, 7. Number NOT divisible by 3.
k) 927: Consider no digits. Number IS divisible by 3.