MathChat

I’ve explained a number of divisibility rules lately, offering tricks to tell if numbers are divisible by 2, 3, 4, 5, 6 and 7.

There is also a trick for divisibility by 8, and that’s what I’d like to explain in this post.

Essentially the trick for 8 is a lot like the trick for 4. If you’d like to refresh your memory on how that trick works, just go here.

The basic idea is that you use the DPM and the DPS, the Divisibility Principle of Multiples and the Divisibility Principle of Sums. My post on the trick for 4 explains both of these principles, so I won’t explain them in depth here.

But I will say that the basic approach we use  is this.

If we’re wondering if a number, call it x, divides into another number, call it y, we do the following.

1)  Break y apart into two addends, call them p and q. We choose these p and q numbers in such a way that we know that x goes into p evenly.

2) Then all we have to do is check to see if x also goes into q. If it does, then x DOES go into y; if x does not go into q, then we can be certain that x does NOT go into y. That is how powerful these two DPM and DPS principles are!

Here’s a quick example of how this works with 4. Suppose we want to know if 4 divides evenly into 252. We know (somehow) that 4 does go into 240. Since 252 = 240 + 12, all we need to do is check to see if 4 also goes into the difference, which is 12. This number, 12, is like the q-number described above. If 4 does go into 12, we can be sure that 4 DOES divide evenly into 252. And if 4 does NOT go into 12, we can be equally sure that 4 does NOT go into 252.

Of course we do know that 4 goes into 12 evenly, so we can be certain that 4 DOES divide evenly into 252.

With 4, we realized that 4 divides evenly into 20 and 100, and those two facts turned out to be really helpful. Knowing that, we were able to subtract multiples of both 20 and 100 from the numbers we were checking out. Doing that, we were able to get a small manageable number (12 in the example), and checking if 4 went into that small number, we figured out if 4 went into the whole number.

With 8, we use the same approach, only now we use the facts that 8 divides evenly into 40 and 200.

The fact that 8 goes into 40 lets us know that we can subtract out multiples of 40:  40, 80, 120, 160, and 200.

The fact that 8 divides into 200 allows us to subtract out the multiples of 200:  200, 400, 600, 800, and 1,000.

And there’s also a bonus to the fact that 8 divides evenly into 1,000. Since that’s true, we know that 8 also divides evenly into all multiples of 1,000. That being the case, we never have to even bother worrying about digits in the thousands place or any place value larger than 1,000. We can confine our search to the number’s final three digits:  the digits in the hundreds, tens, and ones places.

So here’s an example of how we check to see if 8 divides evenly into a number. We’ll check if 8 divides evenly into 736.

1st)  Subtract the nearest (but lesser) multiple of 200. That would be 600. 736 – 600 = 136. We’ve whittled our search down to 136.

2nd)  Subtract the nearest (but lesser) multiple of 40. That would be 120. 136 – 120 = 16.

3rd)  Check to see if 8 goes into the difference. 8 does go into 16 evenly, so 8 DOES divide into 736 evenly. End of story.

Another example:  Does 8 divide evenly into 486.

1st)  Subtract the nearest (but lesser) multiple of 200. That would be 400. 486 – 400 = 86. We’ve now narrowed our search down to 86.

2nd)  Subtract the nearest (but lesser) multiple of 40. That would be 80. 86 – 80 = 6.

3rd)  Check to see if 8 goes into the difference, 6. Of course, 8 does not go into 6, so 8 does NOT divide evenly into 486. End of story.

Example #3:  Does 8 divide evenly into 73,984.

1st)  Disregard the digits greater than the hundreds digit. That means we confine our search to the 984.

2nd)  Subtract the nearest (but lesser) multiple of 200. That would be 800. 984 – 800 = 184. Our search has now narrowed down to 184.

2nd)  Subtract the nearest (but lesser) multiple of 40. That would be 160. 184 – 160 = 24. Our search is now down to just 24.

3rd)  Check to see if 8 goes into the difference, 24. As we know, 8 does divide into 24, so 8 DOES divide evenly into 73,984. End of story, once again.

PRACTICE: Check to see if 8 divides evenly into these numbers.

a)  104
b)  178
c)  472
d)  544
e)  770
f)  853
g)  2,496
h)  5,952
i)  17,038
j)  456,618
k)   2,438,296

ANSWERS:

a)  104:  Divisible by 8
b)  178:  Not divisible by 8
c)  472:  Divisible by 8
d)  544:  Divisible by 8
e)  770:  Not divisible by 8
f)  853:  Not divisible by 8
g)  2,496:  Divisible by 8
h)  5,952:  Divisible by 8
i)  17,038:  Not divisible by 8
j)  456,618:  Not divisible by 8
k)   2,438,288:  Divisible by 8

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