There are many occasions in math problems when it is useful to know if one number is divisible by another. For example, if you can tell at a glance that 3,521 is not divisible by 11, then you will not waste your time trying to divide 3,521 by 11! Or if you need to reduce a fraction such as , it can be very useful to tell at a glance that 252 and 1,080 are both divisible by 9. This can quickly facilitate the reduction of this fraction to , which can then quickly by reduced to when 28 and 120 are both divided by 4.
There are certain slick tricks to tell at a glance if one number is divisible by another. They’re called “Tests for Divisibility.” Everybody knows the simplest test for divisibility. If a number is even (i.e. if its ones digit is 2, 4, 6, 8, or 0) then that number is divisible by 2. It doesn’t matter how big the number is or what the rest of its digits are. This follows from the fact that 10 is divisible by 2, so after every cycle of 10, counting by 2’s starts over with 2, 4, 6, 8. We can use this test to tell at a glance that 135,792 is divisible by 2 but 204,863 is not.
The tests for divisibility by 2, 5, and 10 all have to do with ones digits, so they are very simple, well-known, and intuitive. Tests for divisibility by 3, 6, and 9 are slightly more complicated but also fairly well-known. (They have to do with the sum of digits). If you do some online research, you will probably even find the tests for divisibility by 7 and 11 – important because 7 and 11 are small prime numbers.
In my experience, I have never seen a particularly good test for divisibility by 4, and I’ve never seen a test for divisibility by 8 at all. That bothered me, because 4 and 8 are such primary numbers (powers of 2, less than 10) that my gut told me there must be an easy algorithm for divisibility by these numbers. The only test I’ve ever seen for divisibility by 4 is: “If the last two digits form a multiple of 4, then the number is divisible by 4.” For example, the number 5,128 is divisible by 4 because 28 is divisible by 4. This follows because 100 is divisible by 4, so after every multiple of 100, counting by 4’s starts over with 04, 08, 12, … 88, 92, 96. However, if you don’t know at a glance that 92 is divisible by 4 but 86 is not, then this shortcut won’t help you much with numbers like 592 or 37,086! (For the record, if you want to be a seriously good math student, I do recommend that you memorize the “Extended Times Table” for all products up to 100. That’s a blog for another day).
I knew that the key to divisibility by 4 must lie in the simple fact that 4 goes into 20. Thus, after every multiple of 20, the multiples of 4 cycle through the same pattern of ones digits 4, 8, 2, 6 (such as 24, 28, 32, 36, or 64, 68, 72, 76). Notice that whenever the 1’s digit is divisible by 4, the 10’s digit is even. For instance, 24, 48, and 84 are all divisible by 4. Whenever the 1’s digit is not divisible by 4, the 10’s digit is odd (e.g. 32, 56, or 76).
This rule is still a little complicated, so I learned that I can state it differently, in a way that everyone can understand and memorize:
Test for Divisibility by 4
Double the tens digit and add it to the ones digit.
If your answer is divisible by 4, then so is the number you started with.
Here are some examples:
Question: How can I tell if 156 is divisible by 4?
Answer: Double the 5 and add it to the 6. This gives 10 + 6, which is 16. Since 16 is divisible by 4, it follows that 156 is divisible by 4.
Question: How can I tell if 3,978 is divisible by 4?
Answer: Double the 7 and add to the 8. This gives 14 + 8, which is 22. Since 22 is not divisible by 4, it follows that 3,978 is not divisible by 4.
Frankly, it isn’t hard to just memorize the multiples of 4 up to 100. The real power of this test is that it provides us with the pattern to figure out the test for divisibility by 8. The test is based on the similar fact that 8 goes into 200. Thus, after every multiple of 200, counting by 8’s repeats a cycle such as 208, 216, 224, …, 376, 384, 392. The last two digits always form a multiple of 4, but not necessarily a multiple of 8. Whenever the last two digits form a multiple of 8, the hundreds digit is even. When the last two digits do not form a multiple of 8, the hundreds digit is odd. These facts are summarized in this nice test:
Test for Divisibility by 8
Quadruple the hundreds digit, and then add to the number formed by the tens and ones digits.
If your answer is divisible by 8, then so is the number you started with.
Here are some examples:
Question: How can I tell if 472 is divisible by 8?
Answer: Quadruple the 4 and add to the 72. This results in 16 + 72, which is 88. Since 88 is divisible by 8, it follows that 472 is divisible by 8.
Question: How can I tell if 3,860 is divisible by 8?
Answer: Quadruple the 8 and add to to 60. This results in 32 + 60, which is 92. Since 92 is not divisible by 8, it follows that 3,860 is not divisible by 8.
Question: How can I tell if 2,992 is divisible by 8?
Answer: Quadruple the 9 and add to the 92. This results in 36 + 92 = 128. Uh-oh, a big number! Repeat!
How can I tell if 128 is divisible by 8?
Answer: Quadruple the 1 and add to the 28. The result is 4 + 28, which is 32. OK, this is divisible by 8. Therefore, so is 128, and consequently so is 2,992!!
Technically, these tests can continue indefinitely for powers of 2, but beyond 8 they are no longer simple enough to be helpful. For instance, we could express the test for divisibility by 16 using the same pattern:
Octuple the thousands digit, and then add to the number formed by the last three digits. If your answer is divisible by 16, then so is the number you started with.
Just one last example will illustrate why this doesn’t help much.
Question: How can I tell if 3,464 is divisible by 16?
Answer: Octuple the 3 and add to the 464. This results in 24 + 464, which is 488.
Repeat: How can I tell if 488 is divisible by 16? Octuple the thousands digit (0) and add to the 488. This results in 0 + 488, which is still 488. Unless you know the multiples of 16 up to 1,000 off the top of your head, this isn’t much of a shortcut!