Divisibility by 7: Is there really a trick?

When I talk to people about divisibility, one thing I often hear is:

“There’s not really a trick for 7, is there?”

The question makes a lot of sense. For, of all the one-digit numbers, 7 is in a sense the weirdest digit. It’s odd — and in more ways than one. It does not neatly fit into our base-10 system; it is not half of 10, as is 5. And it is not even next to 0, 1, 5 or 10. It is kind of floating in the midst of the pack, but with no position that makes it distinctive. 7 is just a very weird number. And if you don’t believe me, just look at the decimal expansion of fractions that have 7 as the denominator. 2/7, for example, equals 0.2857142857 …  No other digit from 1 – 9, acting as the denominator, creates such a weird decimal tail (7 digits before it starts to repeat!).

Be that as it may, it’s pretty amazing that there is a “bona fide” trick for figuring out if 7 divides into another number, and I’m going to share it here. But like everything else about 7, even the divisibility trick is weird, so it’s hard to explain it without an example. This being so, I’ll demonstrate this trick using 154 as an example, so you can follow the trick more easily.

The three steps to this divisibility trick:

1st)  Break the number in question into two parts:  a)  the ones digit, and b) the rest of the number, meaning the digits to the left of the ones digit. Just to have a handy way of talking, we’ll call the rest of the number the “leftover.” You read the “leftover” as a number in its own right, reading the digits the standard way, left to right.
For 154, the ones digit is 4. The “leftover” is the number 15.

2nd)   Double the ones digit, and subtract the result from the leftover.
In our example, we double 4, getting 8. Then we subtract 8 from the leftover.  15 – 8 = 7.

3rd)  If the result of the subtraction is either 0 or a multiple of 7 (positive or negative), then the original number IS divisible by 7.
In our example, we see — with amazement (haha) — that the result is 7, which is of course  divisible by 7. So hurray, the original number, 154, is divisible by 7. (Did you really think I’d give an example in which the number is NOT divisible by 7? Have you no faith?)

After all that, you may be forgiven for thinking:  Geez, that trick was so “easy” (haha) … should we really call it a trick?

Of course it is a trick! It’s just not a super-duper-cinchy trick. But actually, if you use it a few times, I think you’ll find it fun and helpful, at least from time to time.

But, just in case you don’t believe me, I’ll offer a more intuitive way to think about divisibility by 7. Just use the two principles I talked about in my post on divisibility by 4.

The two principles are the DPP and the DPS, Divisibility Principle of Products, and the Divisibility Principle of Sums.

Using these tricks means that you do the following thought-steps to test divisibility by 7.

Take a number like 371.

Bear in mind that 7 goes into 35, so it will go into 35o. Subtract 350 from 371. Check out the difference. It is, totally coincidentally (haha), 21, a multiple of 7! So 7 DOES go into 371.

Similarly, to test a number like 529, think about the nearest big multiple of 7;  7 x 7 = 49, so 7 x 70 = 490. Subtract 490 from 529, and you get 39. 7 does NOT go into 39. So fuhgeddaboudit!  7 will NOT go into 529!

Essentially you find the nearest “big” multiple of 7 just below the number in question. You subtract that from the number, and look at the difference to see if 7 goes in. If SO, the original number IS divisible by 7. If NOT, the original number is NOT divisible by 7. That simple.

By now, you might be suspecting that using these DPP and DPS principles will work for any number. At least I am hoping you’re suspecting this. Are you?

If so, good. Because they will work. You can always use this technique — finding a large multiple just below the number in question and subtracting it out — to test for divisibility by any number. True, it is not a super-neat, fast trick, like the trick for 3. But it’s reliable, and with just a little practice, you’ll get quick with it.

In any case, now’s the time to test your new skills with divisibility by 7. See if 7 goes into the following numbers, and if you use the main trick described here, show how you got your answers.

PRACTICE:

a)  91
b)  92
c)  85
d)  84
e)  188
f)  189
g)  336
h)  437
i)   672
j)  763
k)  916
l)   1,491

ANSWERS:

a)  91:  9 – 2 = 7, DIVISIBLE by 7
b)  92:  9 – 4 = 5, NOT divisible by 7
c)  85:  8 – 10 = – 2, NOT divisible by 7
d)  84:  8 – 8 = 0, DIVISIBLE by 7
e)  188:  18 – 16  =  2, NOT divisible by 7
f)  189:  18 – 18 = 0, DIVISIBLE by 7
g)  336:  33 – 12 = 21, DIVISIBLE BY 7
h)  437:  43 – 14 = 29, NOT divisible by 7
i)   672:  67 – 4 = 63, DIVISIBLE by 7
j)  763:  76 – 6 = 70, DIVISIBLE by 7
k)  916:  91 – 12 = 79, NOT divisible by 7
l)   1,491:  149 – 2 = 147, DIVISIBLE by 7

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One Response to “Divisibility by 7: Is there really a trick?”

  1. ZeroSum Ruler says:

    Another awesome post! I’ll use these if and when I teach math again.

    I was hoping you could talk a bit on why we need to double the ones digit….

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