Why Is 3 A Special Denominator In Division?

What do I mean by 3 being a special denominator in division? It is my way of saying that there is something unique about the number: 3 when it occurs as a denominator. Here it is: If the sum of the digits of the numerator is divisible by 3, then the numerator is also divisible by 3. That is, the remainder is zero. For example, to check if 7215 is divisible by 3, I first sum its digits: 7+2+1+5, which gives me 15. I know that 15 is divisible by 3. Therefore, I can say that 7215 is divisible by 3. A quick check on the calculator reveals that 7215/3 = 2405. So, the remainder is indeed zero.

Let’s try the same out for the number 2, shall we? Consider 7242. The sum of the digits is 7+2+4+2, which also gives me 15. But this time, I know that 15 is not divisible by 2. At the same time, I know that 7242 is divisible by 2 (7242/2 = 3621). What about 5? Let us consider 6135. If I sum the digits, I get 6+1+2+5 = 14. I know that 14 is not divisible by 5, yet 6135 is (6135/5 = 1227). So, it seems that 3 is indeed a special number in this regard. At this point, if you share the same human curiosity as me, we have to ask THE question. WHY?! Let’s try to unravel this puzzle further, and see what else we can gain from the insights.

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The Decimal Number System

Before we solve the puzzle of 3’s special property, it helps to take a look at how the decimal number system works. The rightmost number (before the decimal point) occupies the one’s place. The number immediately leftward to it occupies the ten’s place. The number that follows occupies the hundred’s place, and so on. Using this understanding, we could write bigger numbers as the sum of their individual digits multiplied by the place value that they occupy. Here are a few examples that help clarify this:

Why is 3 a Special Denominator?

If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3. We can use the property of the decimal number system that we just reviewed to help us understand 3’s special property. Let us consider the following example: 612. Let us then assign a letter to each digit such that a = 6, b = 1, and c =2. Then we can write down the number 612 as a sum as follows:

We can now rewrite the above equation as follows:

It is quite straightforward that (99*a + 9*b) is divisible by 3 because it involves 9 as a common factor. For the rest of the equation to be divisible by 3, it is a necessary condition for (a + b + c) to be divisible by 3. That is, (a + b + c)/3 gives zero as the remainder. We can scale this approach to bigger numbers, and we’ll see that we always end up with an (a + b + c) term that has to be divisible by 3. This is pretty much why if the sum of the digits of a number is divisible by 3, the number is also divisible by 3. Let’s see what more we can learn from this property next.

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Shuffling Order Does Not Matter

Based on what we have seen so far, we know that 612 is divisible by 3. 612/3 = 204. Now that we have established that a number is divisible by 3 if the sum of its digits is divisible by 3 (in this case, 9), the order in which they occur does not matter. In other words, all combinations of 6,1, and 2 will still be divisible by 3. That is, 612, 261, 126, 621, 216, and 162 are all divisible by 3. This is a pretty neat sub-property to make a note of.

Recursive Application Helps Mental Calculations

Let’s say that you are required to check divisibility by 3 for a large number and that you don’t have access to a calculator. The process that we just followed also works recursively. In simpler terms, we could just keep summing the digits recursively until we arrive at a number small enough for us to compute mentally, and it will still tell us whether the original number is divisible by 3. Here is an example to give you a practical demonstration:

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Further reading that might interest you: Does Division By Zero Really Lead To Infinity? and How To Use Mathematics To Choose A Life Partner?