My last post offered a neat trick for seeing if 3 divides evenly into a number.

In this post, I’ll do the same thing for the number 4.

But my approach will be a bit different in this post. Instead of just presenting the “trick,” I will help us grasp the logic behind the trick by looking at two principles of divisibility. I’m doing this because learning the principles should boost your ability to work — or should I say, play? — with numbers.

First, a question: If a number divides evenly into one number, will it divide evenly into all multiples of that number? Example, given that 6 divides evenly into 30, will 6 divide evenly into the multiples of 30, such as 60, 90, 120, 150, etc. 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.

Second, related question: if a number divides into two other numbers evenly, will it also divide evenly into the sum of those numbers? Check this out with an example, and see if it agree with your mathematical “common sense,” aka “number sense.”

4 goes into both 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.

Test one more example with larger numbers. 9 goes into 90 and 36, right? So does that mean that 9 also must go into 90 + 36, which is 126? Yes again. This idea harmonizes with “number sense,” and it is in fact true. And we will use this soon. We’ll call this the Divisibility Principle of Sums, or just DPS.

To get started thinking about divisibility by 4, let’s consider one nice thing about 4: it divides evenly into a number that ends in 0, the number 20! This is helpful because in our base-10 number system, numbers that end in 0 are “friendly” — they fit into the system neatly.

Using DPM, then, since 4 goes into 20, it goes into all the multiples of 20: 20, 40, 60, 80, and yes, 100! Why is this a big deal? Since 4 goes into 100, we can use DPM again to say 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]

The implication of this is major: if we want to figure out if 4 goes into any whole number, we can ignore all but the last two digits. In other words, to figure out if 4 goes into 5,296 we need only ask: does 4 go into 96. The reason is that we already know that 4 goes into 5,200, and using DPS, if 4 goes into both 5,200 and 96, we can be certain that 4 will go into 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 we can get even more precise.

First ask: before 4 goes into 20, what other numbers does 4 divide into? Simple, 4 goes into 4, 8, 12, and 16.

DPS, we recall, says that if a number, let’s call it n, goes into two other numbers — call them a and b — then n 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 bold numbers below:

20 + 4 = **24**

20 + 8 = **28**

20 + 12 = **32**

20 + 16 = **36**

Big deal, you say, since you already knew this from the times tables. True, but going up one multiple of 20, you can start to see the power of this idea.

Since 4 divides into 40, and into 4, 8, 12 and 16, 4 also goes into the bold numbers:

40 + 4 = **44**

40 + 8 = **48**

40 + 12 = **52**

40 + 16 = **56**

Once again, since 4 divides into 60, and into 4, 8, 12 and 16, 4 also goes into:

60 + 4 = **64**

60 + 8 = **68**

60 + 12 = **72**

60 + 16 = **76**

Using the same pattern, we see that 4 goes into: **80, 84, 88, 92 **and** 96**.

Great, you might say, this shows us a pattern, but not a “trick.”

Where is this 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) **Take 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 number you get by subtracting. If it’s a multiple of 4, then 4 DOES got into the original number. If it is NOT a multiple of 4, then 4 does NOT go into the original number.

Words, words, words, right? Let’s see some examples to give the words some life!

**EXAMPLE 1: **

**Does 4 divide into 58?**

**PROCESS:**

**— Nearest multiple of 20 to 58 is 40.**

— 58 – 40 = 18

— 18 is NOT a multiple of 4, so 4 does NOT divide evenly into 58.

EXAMPLE 2:

**Does 4 divide into 376?**

**PROCESS:**

**— Focus on the last two digits: 76**

**— Nearest multiple of 20 to 76 is 60.**

— 76 – 60 = 16

— 16 IS a multiple of 4, so 4 DOES divide evenly into 376.

**EXAMPLE 3: **

**Does 4 divide into 57,794?**

**PROCESS:**

**— Focus on the last two digits: 94.**

**— Nearest multiple of 20 to 94 is 80.**

— 94 – 80 = 14

— 14 is NOT a multiple of 4, so 4 does NOT divide evenly into 57,794.

Make sense? If so, then you are ready to do some serious divisibility work with 4. Here are some practice problems, and their answers.

**PROBLEMS: Tell 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.

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Written

on February 7, 2013