by Josh Rappaport
In my last two articles, we looked at divisibility tricks for by 2, 3, and 4. For this, the final article on divisibility, we’ll befriend a new number, 6, and think about how divisibility works for it.
I realize this sounds like a narrow topic for an article. But as we go, we’ll discover some general facts about numbers. So hopefully, you and your children will come away learning some larger truths about how numbers work.
Getting back to our new friend, the number 6, the question is: how can we tell if 6 divides evenly into a large number, a number like 372 or 23,478?
The first thing to know is that there’s a general principle that has to do with divisibility. But before I explain that principle, we need to make sure we remember two important math concepts: primes and products.
PRIMES, as you probably recall, are numbers that no numbers go into except for 1 and the number itself. Take, for example, the number 13. The only numbers that divide evenly into 13 are 1 and 13. That means that 13, and all numbers like 13 in this way, are PRIME. FYI: the first prime numbers are: 2, 3, 5, 7, 11, 13, 17 and 19. (Project idea: ask your kids figure out what the next primes are, up to a certain number. Hint: there are 25 primes from 1 – 100. If you want to look this up to see if your kids are right, just Google “primes from 1 – 100.”)
PRODUCT just means the number you get when you multiply two numbers together. So the product of 4 and 2 is 8; the product of 10 and 3 is 30.
So here’s the divisibility principle we’re ready to learn:
If two different prime numbers divide evenly into a number, then the product of those prime numbers ALSO divides evenly into that number.
For example, since the prime numbers 2 and 7 both divide into 42 evenly, that means that the product of 2 and 7, which is 14, should ALSO divide into 42 evenly. And when we check, we see that 14 DOES go into 42 evenly — 3 times, to be exact.
This general principle relates to 6 because 6 IS the product of two primes: 2 and 3.
And this gives us our “trick” for divisibility by 6. To find out if 6 goes into a number evenly, check if both 2 and 3 go into it evenly. If both 2 and 3 DO go into the number evenly, then you can be CERTAIN that 6 goes into the number evenly. On the other hand, if EITHER 2 or 3 DOES NOT go into the number evenly, then you can be equally CERTAIN that 6 DOES NOT go into that number evenly.
Let’s test this out with the number 372. Is 372 divisible by 2? Yes, because 372 is even. Is 372 divisible by 3? Yes, because the sum of 372’s digits is 12, a multiple of 3. So this means that 372 should be divisible by 6. And when we check, we see that 6 DOES go into 372 evenly, 62 times, to be exact.
How about the number 484. Is 484 divisible by 2? Yes, because 484 is even. Is 484 divisible by 3? Uh oh, NO, because the sum of 484’s digits is 16, NOT a multiple of 3. So that means that 484 should NOT be divisible by 6. And when we check, we see that 484 ÷ 6 = 80.666 …, so it is, as we predicted, NOT divisible by 6.
What about a larger number, like 23,478. Is 23,478 divisible by 2? Yes, because 23,478 is even. Is 23,478 divisible by 3? Yes, because the sum of its digits is 24, a multiple of 3. So this means that 23,478 should be divisible by 6. And when we check, we see that 6 DOES go into 23,478 evenly, 3,913 times, to be exact.
What about a situation where a number is not divisible by 6, but you want to tweak it so it becomes divisible by 6. Does our “trick” give us a way to do that?
Let’s explore this with the number 7,348. First, is 7,348 divisible by 2? Yes, because it is even. And is 7,348 divisible by 3? No, because the sum of its digits is 22, which is NOT a multiple of 3. Since 7,348 is NOT divisible by 3, it will not be divisible by 6. Boo hoo, right?
But let’s try to work with this. We’ll keep its final digit 8 so it stays even. But we’ll change it so that its digits add up to a multiple of 3. Right now the digits add up to 22. If we could just increase that sum by 2, they’ll add up to 24, which IS a multiple of 3. To do this, let’s just add 2 to the first digit, 7, boosting that 7 up to a 9, which gives us the tweaked number 9,348. Now we know that 9,348 is divisible by both 2 and 3, so it should be divisible by 6, if our theory is right.
Let’s test it out. 9,348 ÷ 6 = (drumroll, please) … 1,558. So 9,348 IS divisible by 6, as we predicted. So once again, the “trick” works.
Isn’t that cool? I mean, math is so certain! And when you know a general mathematical rule, you can use it again and again, and it will ALWAYS work perfectly. I don’t see how anyone could be unimpressed with this. Of course, I’m extremely biased since I think about numbers every day, but I do hope that some of my enthusiasm is rubbing off, even if just a tiny bit.
In any case, this ends our first round of studying divisibility. I hope that you enjoyed it and learned a few interesting things along the way.
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
In my last two articles, we looked at divisibility tricks for by 2, 3, and 4. For this, the final article on divisibility, we’ll befriend a new number, 6, and think about how divisibility works for it.
I realize this sounds like a narrow topic for an article. But as we go, we’ll discover some general facts about numbers. So hopefully, you and your children will come away learning some larger truths about how numbers work.
Getting back to our new friend, the number 6, the question is: how can we tell if 6 divides evenly into a large number, a number like 372 or 23,478?
The first thing to know is that there’s a general principle that has to do with divisibility. But before I explain that principle, we need to make sure we remember two important math concepts: primes and products.
PRIMES, as you probably recall, are numbers that no numbers go into except for 1 and the number itself. Take, for example, the number 13. The only numbers that divide evenly into 13 are 1 and 13. That means that 13, and all numbers like 13 in this way, are PRIME. FYI: the first prime numbers are: 2, 3, 5, 7, 11, 13, 17 and 19. (Project idea: ask your kids figure out what the next primes are, up to a certain number. Hint: there are 25 primes from 1 – 100. If you want to look this up to see if your kids are right, just Google “primes from 1 – 100.”)
PRODUCT just means the number you get when you multiply two numbers together. So the product of 4 and 2 is 8; the product of 10 and 3 is 30.
So here’s the divisibility principle we’re ready to learn:
If two different prime numbers divide evenly into a number, then the product of those prime numbers ALSO divides evenly into that number.
For example, since the prime numbers 2 and 7 both divide into 42 evenly, that means that the product of 2 and 7, which is 14, should ALSO divide into 42 evenly. And when we check, we see that 14 DOES go into 42 evenly — 3 times, to be exact.
This general principle relates to 6 because 6 IS the product of two primes: 2 and 3.
And this gives us our “trick” for divisibility by 6. To find out if 6 goes into a number evenly, check if both 2 and 3 go into it evenly. If both 2 and 3 DO go into the number evenly, then you can be CERTAIN that 6 goes into the number evenly. On the other hand, if EITHER 2 or 3 DOES NOT go into the number evenly, then you can be equally CERTAIN that 6 DOES NOT go into that number evenly.
Let’s test this out with the number 372. Is 372 divisible by 2? Yes, because 372 is even. Is 372 divisible by 3? Yes, because the sum of 372’s digits is 12, a multiple of 3. So this means that 372 should be divisible by 6. And when we check, we see that 6 DOES go into 372 evenly, 62 times, to be exact.
How about the number 484. Is 484 divisible by 2? Yes, because 484 is even. Is 484 divisible by 3? Uh oh, NO, because the sum of 484’s digits is 16, NOT a multiple of 3. So that means that 484 should NOT be divisible by 6. And when we check, we see that 484 ÷ 6 = 80.666 …, so it is, as we predicted, NOT divisible by 6.
What about a larger number, like 23,478. Is 23,478 divisible by 2? Yes, because 23,478 is even. Is 23,478 divisible by 3? Yes, because the sum of its digits is 24, a multiple of 3. So this means that 23,478 should be divisible by 6. And when we check, we see that 6 DOES go into 23,478 evenly, 3,913 times, to be exact.
What about a situation where a number is not divisible by 6, but you want to tweak it so it becomes divisible by 6. Does our “trick” give us a way to do that?
Let’s explore this with the number 7,348. First, is 7,348 divisible by 2? Yes, because it is even. And is 7,348 divisible by 3? No, because the sum of its digits is 22, which is NOT a multiple of 3. Since 7,348 is NOT divisible by 3, it will not be divisible by 6. Boo hoo, right?
But let’s try to work with this. We’ll keep its final digit 8 so it stays even. But we’ll change it so that its digits add up to a multiple of 3. Right now the digits add up to 22. If we could just increase that sum by 2, they’ll add up to 24, which IS a multiple of 3. To do this, let’s just add 2 to the first digit, 7, boosting that 7 up to a 9, which gives us the tweaked number 9,348. Now we know that 9,348 is divisible by both 2 and 3, so it should be divisible by 6, if our theory is right.
Let’s test it out. 9,348 ÷ 6 = (drumroll, please) … 1,558. So 9,348 IS divisible by 6, as we predicted. So once again, the “trick” works.
Isn’t that cool? I mean, math is so certain! And when you know a general mathematical rule, you can use it again and again, and it will ALWAYS work perfectly. I don’t see how anyone could be unimpressed with this. Of course, I’m extremely biased since I think about numbers every day, but I do hope that some of my enthusiasm is rubbing off, even if just a tiny bit.
In any case, this ends our first round of studying divisibility. I hope that you enjoyed it and learned a few interesting things along the way.
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
