## Monday, August 28, 2017

### Proof that the sum of digits of any multiple of 3 (or 9) is itself a multiple of 3 (or 9)

Take any multiple of 3, such as 327. You can verify that it really is a multiple of 3 by adding together all its digits, that is 3+2+7=12, and checking if the sum is itself a multiple of 3. 12 is a multiple of 3, so 327 must also be a multiple of 3. Since the sum of digits will be a much smaller number than the original number, it will be easier to determine if it is a multiple of 3 or not. Of course you can then take the sum of digits and find its sum of digits again repeatedly until it's a single digit number: 3, 6, 9. The same thing is true of multiples of 9 but instead the sum of digits will be a multiple of 9.

This is the proof that this works for every multiple of 3:

Start with a number with $n$ digits, such as the 3 digit number 327. We'll call each of these digits $a_i$, starting from $a_{n-1}$ for the first digit and ending with $a_0$ for the last digit. So 327 has $a_2 = 3$, $a_1 = 2$, and $a_0 = 7$.

Our number is equal to the following:
$$a_{n-1} 10^{n-1} + a_{n-2} 10^{n-2} + \ldots + a_1 10^1 + a_0 10^0$$
This is just the hundreds, tens, units formulation of numbers. For example:
$$327 = 3 \times 10^2 + 2 \times 10^1 + 7 \times 10^0 = 3 \times 100 + 2 \times 10 + 7 \times 1 = 327$$

Now the trick is to replace each $10^x$ with $999...+1$. For example $100 = 99+1$.

$$\begin{eqnarray} a_{n-1} 10^{n-1} + a_{n-2} 10^{n-2} + \ldots + a_1 10^1 + a_0 10^0 &=& a_{n-1} \left(999... + 1\right) + a_{n-2} \left(99... + 1\right) + \ldots + a_1 \left(9 + 1\right) + a_0 \left(0 + 1\right) \\ &=& \left(a_{n-1} 999... + a_{n-1}\right) + \left(a_{n-2} 99... + a_{n-2}\right) + \ldots + \left(a_1 9 + a_1\right) + \left(a_0 0 + a_0\right) \\ &=& \left(a_{n-1} 999... + a_{n-2} 99... + \ldots + a_1 9 + a_0 0\right) + \left(a_{n-1} + a_{n-2} + \ldots + a_1 + a_0\right) \\ \end{eqnarray}$$

Now we'll make some terms switch places in the equation:
$$\begin{eqnarray} a_{n-1} + a_{n-2} + \ldots + a_1 + a_0 &=& \left(a_{n-1} 10^{n-1} + a_{n-2} 10^{n-2} + \ldots + a_1 10^1 + a_0 10^0\right) - \left(a_{n-1} 999... + a_{n-2} 99... + \ldots + a_1 9 + a_0 0\right) \\ &=& \left(a_{n-1} 10^{n-1} + a_{n-2} 10^{n-2} + \ldots + a_1 10^1 + a_0 10^0\right) - \left(a_{n-1} 999... + a_{n-2} 99... + \ldots + a_1 9\right) \end{eqnarray}$$

Notice that in the last line above we have eliminate the term $a_0 0$ because it's a multiplication by 0.

Now, if our original number is a multiple of 3, then it must be that the right hand side at the end of the above equation is a multiple of 3. Any string of 9s is always a multiple of 3 since $999\ldots = 3 \times 3 \times 111\ldots$. It is also a multiple of 9, but we'll get to that later. This means that the two bracketed terms in the last line of the above equation are both multiples of 3 because:
• $a_{n-1} 10^{n-1} + a_{n-2} 10^{n-2} + \ldots + a_1 10^1 + a_0 10^0$: is a multiple of 3 by definition (we said that the number would be one).
• $a_{n-1} 999... + a_{n-2} 99... + \ldots + a_1 9$: is a sum of numbers multiplied by strings of 9s, which means that we can factor out the common factor 3.

The difference of two multiples of 3 is itself a multiple of 3. Therefore the left hand side must also be a multiple of 3 (since the two sides are equal), and the left hand side just happens to be:

$$a_{n-1} + a_{n-2} + \ldots + a_1 + a_0$$

which is the sum of all digits. Hence for any number that is a multiple of 3, the sum of its digits must also be a multiple of 3.

Until now we have only shown that any multiple of 3 will have the sum of its digits also be a multiple of 3. But can a number which is not a multiple of 3 coincidentally have the sum of its digits be a multiple of 3? No, because otherwise it would imply that a non-multiple of 3 minus a multiple of 3 is a multiple of 3. The second term in the subtraction in the right hand side of the above derivation is always going to be a multiple of 3, but in order for the whole of the right hand side to be a multiple of 3 you will need both terms being a multiple of 3. So you can rest your mind that checking if a large number is a multiple of 3 can be as simple as checking if the sum of its digits is a multiple of 3. And if that sum is still too big you can check if the sum of its digits are a multiple of 3 and so on, because you're always just reusing the same test each time.

What about multiples of 9? As already mentioned above, a string of 9s is also always a multiple of 9. So the second term in the subtraction above is also always a multiple of 9. Hence if both terms of the subtraction are multiples of 9, then the sum of digits is going to be a multiple of 9.