by Josh Rappaport
My last article offered a quick trick for finding out if 3 divides evenly into a number. If that made you hungry for more divisibility tricks, you’re in luck.
In this article, I'll provide a similar, fast divisibility trick for the number 4.
This month, instead of presenting only the "trick," I’ll also help us grasp the logic behind the trick by pointing out two principles of divisibility. I'm doing this because understanding such principles can boost your ability to work with numbers, just as understanding how an engine works boosts a person’s ability to repair a broken-down car.
First, an introductory question: if a number divides evenly into another number, will it divide evenly into ALL MULTIPLES of that other number? For example, knowing that 6 divides evenly into 12, would we know with absolute certainty that 6 divides evenly into all multiples of 12: 24, 36, 48, 60, etc.
You probably want to say yes, but that business about “absolute certainty” is a bit intimidating. But don’t fret. The answer is YES. This is a basic principle of divisibility, and we'll call it the Divisibility Principle of Multiples, or just DPM, for short.
Now for a related question: if a number divides evenly into two other numbers, will it also divide evenly into the SUM of those numbers?
Let’s explore this through an example.
We know that 4 divides evenly into 20 and 8, right? So does that mean that 4 goes into the sum of 20 and 8, namely 28? Well, yes, 4 does go into 28 evenly, seven times in fact.
To test this idea further, let’s check it out with larger numbers. We know that 9 divides evenly into 90 and 36, right? So does that mean that 9 also absolutely must divide evenly into 90 + 36, which is 126? Yes again! And this idea is true in general as well. To abbreviate it, we’ll call it the Divisibility Principle of Sums, the DPS.
So now, to start thinking about divisibility by 4, let's consider one nice thing about 4 that relates to both the DPM and DPS: 4 divides evenly into a “friendly” number, a number that ends in 0, the wonderful number 20! This is helpful because in our base-10 number system, numbers that end in 0 fit the system neatly.
Using the DPM, since 4 goes into 20, it goes into all multiples of 20: 20, 40, 60, 80, and also the supremely beautiful number: 100! Why is this a big deal? Since 4 goes into 100, we can use the DPM to see that 4 goes into all multiples of 100: 200; 300; 400; ... 700; 1,300; 2,300, ... we can even be certain that 4 goes into 6,235,700 since this is a multiple of 100 [100 x 62,357 = 6,235,700]. In fact, this means that 4 divides evenly into any number that ends in -00.
The implication of this is HUGE: if we want to know whether or not 4 divides evenly into any whole number, we can ignore all but the last two digits. In other words, to figure out if 4 divides evenly into 5,296 we need only ask: does 4 go into 96. The reason? We already know that 4 goes into 5,200. So using DPS, if 4 goes into both 5,200 and 96, we can be absolutely sure that 4 divides evenly into their sum: 5,296.
So we now have the first part of our trick for 4: To find out if 4 goes into any number, LOOK ONLY AT THE LAST TWO DIGITS.
That's a great start. But can get more precise?
First ask what numbers 4 goes into that are less than 20. Simple. 4 goes into 4, 8, 12 and 16.
DPS, we recall, assures us that if any number, let's call it n, goes into two other numbers — call them a and b — then n also goes into their sum:
(a + b).
We can use this idea right here. Since 4 divides into 20, and it also divides into 4, 8, 12 and 16, DPS guarantees that 4 also goes into the following sums: 24 (since 24 = 20 + 4); 28 (since 28 = 20 + 8); 32 (since 32 = 20 + 12); and 36 (since 36 = 20 + 16).
Big deal, you say (perhaps annoyed), pointing out that you already knew that 24, 28, 32 and 36 are divisible by 4. Of course knew this. But if we rise by one multiple of 20, you’ll start to see the power of this principle.
Since 4 divides into 40 (as well as into 4, 8, 12 and 16), this means we can be certain that 4 also goes into these sums: 44 (since 44 = 40 + 4); 48 (since 48 = 40 + 8); 52 (since 52 = 40 + 12); and 56 (since 56 = 40 + 16).
Once again, since 4 divides evenly into 60 (as well as into 4, 8, 12 and 16), 4 also goes into: 64, 68, 72, and 76.
And finally, using the same idea, we can be certain that since 4 divides evenly into 80, it also goes into 84, 88, 92 and 96.
Great, you might say, this shows us a pattern, but not a "trick." Where is the long-promised trick?
What we need to realize is that the pattern leads to a trick. For the trick, here's what you do:
1st) Look only at the two digits at the end of any whole number.
2nd) Find the lesser but nearest multiple of 20, and subtract it from the two-digit number.
3rd) Look at the difference you get by subtracting. If it's a multiple of 4, then 4 DOES go into the original number. If it is NOT a multiple of 4, then 4 does NOT go into the original number.
But now, let's see some examples to bring this process to life, right?
EXAMPLE 1: Does 4 divide evenly into 58?
PROCESS: Nearest multiple of 20 to 58 is 40. 58 – 40 = 18, which is NOT a multiple of 4, so 4 does NOT divide evenly into 58.
EXAMPLE 2: Does 4 divide evenly into 376?
PROCESS: Focus on only the last two digits: 76. Nearest multiple of 20 to 76 is 60. 76 – 60 = 16, a multiple of 4, so 4 DOES divide evenly into 376.
EXAMPLE 3: Does 4 divide evenly into 57,794?
PROCESS: The last two digits are 94. Nearest multiple of 20 is 80. 94 – 80 = 14, which is NOT a multiple of 4, so 4 does NOT divide evenly into 57,794.
Make sense? If so, then you and your kiddos are ready to do some serious divisibility work with 4. Here are some practice problems, along with their answers.
PROBLEMS: Find out if 4 divides evenly into the following numbers.
a) 74
b) 92
c) 354
d) 768
e) 1,596
f) 3,390
g) 52,472
h) 831,062
i) 973,236
j) 17,531,958
ANSWERS:
a) 74: 74 – 60 = 14. 4 does NOT divide evenly into 74. b) 92: 92 – 80 = 12. 4 DOES divide evenly into 92. c) 354: 54 – 40 = 14. 4 does NOT divide evenly into 354. d) 768: 68 – 60 = 8. 4 DOES divide evenly into 768. e) 1,596: 96 – 80 = 16. 4 DOES divide evenly into 1,596.
f) 3,390: 90 – 80 = 10. 4 does NOT divide evenly into 3,390. g) 52,472: 72 – 60 = 12. 4 DOES divide evenly into 52,472.
h) 831,062: 62 – 60 = 2. 4 does NOT divide evenly into
831, 062. i) 973,236: 36 – 20 = 16. 4 DOES divide evenly into 973,236. j) 17,531,958: 58 – 40 = 18. 4 does NOT divide evenly int0 7,531,958.
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
My last article offered a quick trick for finding out if 3 divides evenly into a number. If that made you hungry for more divisibility tricks, you’re in luck.
In this article, I'll provide a similar, fast divisibility trick for the number 4.
This month, instead of presenting only the "trick," I’ll also help us grasp the logic behind the trick by pointing out two principles of divisibility. I'm doing this because understanding such principles can boost your ability to work with numbers, just as understanding how an engine works boosts a person’s ability to repair a broken-down car.
First, an introductory question: if a number divides evenly into another number, will it divide evenly into ALL MULTIPLES of that other number? For example, knowing that 6 divides evenly into 12, would we know with absolute certainty that 6 divides evenly into all multiples of 12: 24, 36, 48, 60, etc.
You probably want to say yes, but that business about “absolute certainty” is a bit intimidating. But don’t fret. The answer is YES. This is a basic principle of divisibility, and we'll call it the Divisibility Principle of Multiples, or just DPM, for short.
Now for a related question: if a number divides evenly into two other numbers, will it also divide evenly into the SUM of those numbers?
Let’s explore this through an example.
We know that 4 divides evenly into 20 and 8, right? So does that mean that 4 goes into the sum of 20 and 8, namely 28? Well, yes, 4 does go into 28 evenly, seven times in fact.
To test this idea further, let’s check it out with larger numbers. We know that 9 divides evenly into 90 and 36, right? So does that mean that 9 also absolutely must divide evenly into 90 + 36, which is 126? Yes again! And this idea is true in general as well. To abbreviate it, we’ll call it the Divisibility Principle of Sums, the DPS.
So now, to start thinking about divisibility by 4, let's consider one nice thing about 4 that relates to both the DPM and DPS: 4 divides evenly into a “friendly” number, a number that ends in 0, the wonderful number 20! This is helpful because in our base-10 number system, numbers that end in 0 fit the system neatly.
Using the DPM, since 4 goes into 20, it goes into all multiples of 20: 20, 40, 60, 80, and also the supremely beautiful number: 100! Why is this a big deal? Since 4 goes into 100, we can use the DPM to see that 4 goes into all multiples of 100: 200; 300; 400; ... 700; 1,300; 2,300, ... we can even be certain that 4 goes into 6,235,700 since this is a multiple of 100 [100 x 62,357 = 6,235,700]. In fact, this means that 4 divides evenly into any number that ends in -00.
The implication of this is HUGE: if we want to know whether or not 4 divides evenly into any whole number, we can ignore all but the last two digits. In other words, to figure out if 4 divides evenly into 5,296 we need only ask: does 4 go into 96. The reason? We already know that 4 goes into 5,200. So using DPS, if 4 goes into both 5,200 and 96, we can be absolutely sure that 4 divides evenly into their sum: 5,296.
So we now have the first part of our trick for 4: To find out if 4 goes into any number, LOOK ONLY AT THE LAST TWO DIGITS.
That's a great start. But can get more precise?
First ask what numbers 4 goes into that are less than 20. Simple. 4 goes into 4, 8, 12 and 16.
DPS, we recall, assures us that if any number, let's call it n, goes into two other numbers — call them a and b — then n also goes into their sum:
(a + b).
We can use this idea right here. Since 4 divides into 20, and it also divides into 4, 8, 12 and 16, DPS guarantees that 4 also goes into the following sums: 24 (since 24 = 20 + 4); 28 (since 28 = 20 + 8); 32 (since 32 = 20 + 12); and 36 (since 36 = 20 + 16).
Big deal, you say (perhaps annoyed), pointing out that you already knew that 24, 28, 32 and 36 are divisible by 4. Of course knew this. But if we rise by one multiple of 20, you’ll start to see the power of this principle.
Since 4 divides into 40 (as well as into 4, 8, 12 and 16), this means we can be certain that 4 also goes into these sums: 44 (since 44 = 40 + 4); 48 (since 48 = 40 + 8); 52 (since 52 = 40 + 12); and 56 (since 56 = 40 + 16).
Once again, since 4 divides evenly into 60 (as well as into 4, 8, 12 and 16), 4 also goes into: 64, 68, 72, and 76.
And finally, using the same idea, we can be certain that since 4 divides evenly into 80, it also goes into 84, 88, 92 and 96.
Great, you might say, this shows us a pattern, but not a "trick." Where is the long-promised trick?
What we need to realize is that the pattern leads to a trick. For the trick, here's what you do:
1st) Look only at the two digits at the end of any whole number.
2nd) Find the lesser but nearest multiple of 20, and subtract it from the two-digit number.
3rd) Look at the difference you get by subtracting. If it's a multiple of 4, then 4 DOES go into the original number. If it is NOT a multiple of 4, then 4 does NOT go into the original number.
But now, let's see some examples to bring this process to life, right?
EXAMPLE 1: Does 4 divide evenly into 58?
PROCESS: Nearest multiple of 20 to 58 is 40. 58 – 40 = 18, which is NOT a multiple of 4, so 4 does NOT divide evenly into 58.
EXAMPLE 2: Does 4 divide evenly into 376?
PROCESS: Focus on only the last two digits: 76. Nearest multiple of 20 to 76 is 60. 76 – 60 = 16, a multiple of 4, so 4 DOES divide evenly into 376.
EXAMPLE 3: Does 4 divide evenly into 57,794?
PROCESS: The last two digits are 94. Nearest multiple of 20 is 80. 94 – 80 = 14, which is NOT a multiple of 4, so 4 does NOT divide evenly into 57,794.
Make sense? If so, then you and your kiddos are ready to do some serious divisibility work with 4. Here are some practice problems, along with their answers.
PROBLEMS: Find out if 4 divides evenly into the following numbers.
a) 74
b) 92
c) 354
d) 768
e) 1,596
f) 3,390
g) 52,472
h) 831,062
i) 973,236
j) 17,531,958
ANSWERS:
a) 74: 74 – 60 = 14. 4 does NOT divide evenly into 74. b) 92: 92 – 80 = 12. 4 DOES divide evenly into 92. c) 354: 54 – 40 = 14. 4 does NOT divide evenly into 354. d) 768: 68 – 60 = 8. 4 DOES divide evenly into 768. e) 1,596: 96 – 80 = 16. 4 DOES divide evenly into 1,596.
f) 3,390: 90 – 80 = 10. 4 does NOT divide evenly into 3,390. g) 52,472: 72 – 60 = 12. 4 DOES divide evenly into 52,472.
h) 831,062: 62 – 60 = 2. 4 does NOT divide evenly into
831, 062. i) 973,236: 36 – 20 = 16. 4 DOES divide evenly into 973,236. j) 17,531,958: 58 – 40 = 18. 4 does NOT divide evenly int0 7,531,958.
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
