by Josh Rappaport
Quick: Which English word has 12 letters (more than Mississippi), almost half of which are the letter "i"?
The word "d-I-v-I-s-I-b-I-l-I-t-y" — which just happens to be a great thing to know if you're doing any branch of math.
But what exactly is “divisibility.” In essence, divisibility is the art of quickly knowing whether or not one number divides evenly into another number —without a calculator.
For example, if you understand divisibility by 6, you can quickly tell whether or not 6 divides evenly into 4,332 (it does) or into 9,846 (it doesn’t).
Similarly, if you grasp divisibility by 9, you can tell at a glance whether or not 9 divides evenly into 3,033 (yep!) or into 52,016 (nope!).
At this point, many readers will have a simple question: who cares?!
Good question! It turns out that learning the rules of divisibility is actually a huge time-saver. (Here’s where the kids get interested!) Once children know the rules of divisibility, ALL of the following math operations become MUCH quicker and easier:
— simplifying fractions
— multiplying fractions
— dividing fractions
— adding and subtracting fractions
— finding the GCF (greatest common factor)
— finding the LCM (least common multiple)
— simplifying ratios
— solving proportions
— factoring algebraic expressions
— factoring quadratic trinomials
I’ll explain how divisibility helps with these operations in future posts.
Even if you’re a smarty-pants (just joshing!) and you already know the divisibility trick for 3, don't skip this article. For after I show how this rule is usually presented, I'll share a couple of “superhero” tricks that most people have never heard of — tricks that make the basic trick even quicker and easier. Then you’ll be a true smarty-pants!
So without further ado, here's the basic divisibility rule for 3.
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 DOES divide evenly into the original number. But 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 evenly into the original number.
Example A: Consider 311.
Add its 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 its 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 this same level of certainty?!
SUPERHERO TRICK #1:
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.
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 b) 16,037 c) 972,132
ANSWERS:
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.
SUPERHERO TRICK #2:
Just as you can disregard individual digits that are 0, 3, 6, and 9, you 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, you can disregard both those digits, since they add up to 9. And if a number has an 8 and a 4, you can disregard both of those digits, 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) 51,927 b) 62,497 c) 102,386
ANSWERS:
a) 51,954: Disregard 5 & 1 (since they add up to 6); disregard the 9; disregard the 2 & 7 (since they add up to 9). So number is DIVISIBLE by 3. [NOTE: If you can disregard all digits, 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 Superhero Tricks?
Using the rule and Superhero Tricks, determine which numbers you need to consider, then decide whether or not 3 divides into these numbers.
a) 47
b) 915
c) 4,316
d) 84,063
e) 25,172
f) 367,492
g) 5,648
h) 12,039
i) 79
j) 617
k) 924
ANSWERS:
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.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
Quick: Which English word has 12 letters (more than Mississippi), almost half of which are the letter "i"?
The word "d-I-v-I-s-I-b-I-l-I-t-y" — which just happens to be a great thing to know if you're doing any branch of math.
But what exactly is “divisibility.” In essence, divisibility is the art of quickly knowing whether or not one number divides evenly into another number —without a calculator.
For example, if you understand divisibility by 6, you can quickly tell whether or not 6 divides evenly into 4,332 (it does) or into 9,846 (it doesn’t).
Similarly, if you grasp divisibility by 9, you can tell at a glance whether or not 9 divides evenly into 3,033 (yep!) or into 52,016 (nope!).
At this point, many readers will have a simple question: who cares?!
Good question! It turns out that learning the rules of divisibility is actually a huge time-saver. (Here’s where the kids get interested!) Once children know the rules of divisibility, ALL of the following math operations become MUCH quicker and easier:
— simplifying fractions
— multiplying fractions
— dividing fractions
— adding and subtracting fractions
— finding the GCF (greatest common factor)
— finding the LCM (least common multiple)
— simplifying ratios
— solving proportions
— factoring algebraic expressions
— factoring quadratic trinomials
I’ll explain how divisibility helps with these operations in future posts.
Even if you’re a smarty-pants (just joshing!) and you already know the divisibility trick for 3, don't skip this article. For after I show how this rule is usually presented, I'll share a couple of “superhero” tricks that most people have never heard of — tricks that make the basic trick even quicker and easier. Then you’ll be a true smarty-pants!
So without further ado, here's the basic divisibility rule for 3.
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 DOES divide evenly into the original number. But 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 evenly into the original number.
Example A: Consider 311.
Add its 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 its 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 this same level of certainty?!
SUPERHERO TRICK #1:
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.
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 b) 16,037 c) 972,132
ANSWERS:
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.
SUPERHERO TRICK #2:
Just as you can disregard individual digits that are 0, 3, 6, and 9, you 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, you can disregard both those digits, since they add up to 9. And if a number has an 8 and a 4, you can disregard both of those digits, 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) 51,927 b) 62,497 c) 102,386
ANSWERS:
a) 51,954: Disregard 5 & 1 (since they add up to 6); disregard the 9; disregard the 2 & 7 (since they add up to 9). So number is DIVISIBLE by 3. [NOTE: If you can disregard all digits, 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 Superhero Tricks?
Using the rule and Superhero Tricks, determine which numbers you need to consider, then decide whether or not 3 divides into these numbers.
a) 47
b) 915
c) 4,316
d) 84,063
e) 25,172
f) 367,492
g) 5,648
h) 12,039
i) 79
j) 617
k) 924
ANSWERS:
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.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
