A faster divisibility rule for $7$ (and $13$)

I occasionally teach evening online classes for Art of Problem Solving (artofproblemsolving.com, a fantastic resource for high school students interested in learning mathematics), and one of the lessons I taught recently was on fast mental arithmetic tricks for testing divisibility by various small integers.

How do you tell if a number is divisible by $2$? Easy: look at the last digit. If that digit is even, the whole number is divisible by $2$, otherwise it’s not.

How do you tell if a number is divisible by $3$? Easy: Add up the digits and see if the sum is divisible by $3$. For instance, $1642$ is not divisible by $3$ because $1+6+4+2=13$ is not, but $1644$ is because $1+6+4+4=15$ is divisible by $3$.

The list goes on; there are nice, well-known divisibility rules for $4$, $5$, $6$, $8$, $9$, $10$, $11$, and $12$. But $7$ turns out to be rather annoying. Many of the existing texts on this subject use a recursive rule that goes something like this: Cross off the last digit, double that digit, and subtract it from the number that remains. Then keep doing that until you get a small number and see if the result is divisible by $7$.

Ouch.

Not only is this method somewhat cumbersome, the small number you get in the end does not tell you the remainder mod $7$ (see this post for an introduction to modular arithmetic), only whether or not that remainder is $0$. But it turns out that one can derive a different method along the same lines as the divisibility rule for $3$. (See this Wikipedia article for many, many divisibility rules!)

First, let’s recall the proof of the rule for $3$. Given a number in base ten, like $1642$, we can write it out as a sum of powers of $10$:

$$1642=1\cdot 10^3+6\cdot 10^2+4\cdot 10+2$$
Notice that $10$ has a remainder of $1$ when divided by $3$, so when looking at remainders mod $3$, we can replace all these $10$’s in the expression above with $1$’s:
$$1642\equiv 1+6+4+2 \pmod{3}$$ and we have recovered the sum of the digits expression.

Similarly, for mod $7$, we can write things out in terms of powers of $10$, and reduce these powers modulo $7$. The number $10$ itself is $3$ mod $7$, then $100\equiv 10^2\equiv 3^2$ is $2$ mod $7$. The next remainders, for $10^3,10^4, 10^5$ are $6,4,5$, and then the remainder when $10^6$ is divided by $7$ is $1$ again, wrapping around to the same value as $10^0$. Writing this out in a table, we see a pattern start to emerge:

$$\begin{array}{cccccccccc}
n & 1 & 10 & 10^2 & 10^3 & 10^4 & 10^5 & 10^6 & 10^7 & 10^8 \\
n \bmod 7 & 1 & 3 & 2 & 6 & 4 & 5 & 1 & 3 & 2
\end{array}$$

Indeed, the remainders $1,3,2,6,4,5$ repeat every six steps, since at each step we are simply multiplying the previous remainder by $10$ and taking the result mod $7$. But we can make this pattern even simpler by replacing $6$ with $-1$, $4$ with $-3$, and $5$ with $-2$, which still have the same remainders mod $7$ (they differ by $7$ from the originals):

$$\begin{array}{cccccccccc}
n & 1 & 10 & 10^2 & 10^3 & 10^4 & 10^5 & 10^6 & 10^7 & 10^8 \\
n \bmod 7 & 1 & 3 & 2 & -1 & -3 & -2 & 1 & 3 & 2
\end{array}$$

So, the pattern simply continues $1,3,2,-1,-3,-2,1,3,2,-1,-3,-2,\ldots$. We can therefore replace the powers of $10$ in a base ten expansion by these smaller digits to compute the number mod $7$. For instance, $$1642=10^3+6\cdot 10^2+4\cdot 10+2\equiv -1+6\cdot 2+4\cdot 3+2=25\equiv 4\pmod 7,$$ so $1642$ has a remainder of $4$ when divided by $7$.

As a larger example, starting with the number $133251140379$, we can make a simple table with the pattern $1,3,2,-1,-3,-2,\ldots$ underneath the digits from right to left (we reverse the direction since the powers of $10$ increase to the left):

$$\begin{array}{cccccccccccc}
1 & 3 & 3 & 2 & 5 & 1 & 1 & 4 & 0 & 3 & 7 & 9 \\
-2 & -3& -1& 2 & 3 & 1 & -2& -3& -1& 2 & 3 & 1
\end{array}$$

Then we can simply multiply the pairs of numbers in column together (and take the results mod 7) and add the results mod $7$. We can write each product below the pairs:

$$\begin{array}{cccccccccccc}
1 & 3 & 3 & 2 & 5 & 1 & 1 & 4 & 0 & 3 & 7 & 9 \\
-2 & -3& -1& 2 & 3 & 1 & -2& -3& -1& 2 & 3 & 1 \\\hline
-2 & -2& -3& 4 & 1 & 1 & -2& -5& 0 & 6 & 0 & 2
\end{array}$$

(Notice that in the computation of the products we sometimes automatically take the result mod $7$: for instance, $5\cdot 3=15\equiv 1\pmod 7$, so under the fifth column we simply write $1$.) Finally, the sum of the numbers in the bottom row is $-2-2-3+4+1+1-2-5+6+2=0$, so $133251140379$ is indeed divisible by $7$.

I tested this method by hand on a few large numbers and compared my calculation time to the iterative subtraction method, and it saves a good chunk of time. The particular number above took me 35 seconds to check with the $1,3,2$-pattern method, and 69 seconds to check with the double-and-subtract rule. Try it out, and let me know in the comments whether you find this method easier!

Addendum: A similar rule for $13$

While this method may become more cumbersome for larger moduli (which will have a longer pattern to memorize), it turns out that powers of $10$ mod $13$ also have an easily-memorizable repeating pattern:

$$\begin{array}{cccccccccc}
n & 1 & 10 & 10^2 & 10^3 & 10^4 & 10^5 & 10^6 & 10^7 & 10^8 \\
n \bmod 13 & 1 & -3 & -4 & -1 & 3 & 4 & 1 & -3 & -4
\end{array}$$

The pattern is $1,-3,-4,-1,3,4$ repeating, which can be memorized by taking the length $3$ string $1,3,4$ and then writing negative signs on every other block of three entries starting at the tens digit. Let’s try it out on our example number above.

$$\begin{array}{cccccccccccc}
1 & 3 & 3 & 2 & 5 & 1 & 1 & 4 & 0 & 3 & 7 & 9 \\
4 & 3 &-1 &-4 &-3 & 1 & 4 & 3 & -1&-4 &-3 & 1 \\\hline
4 &-4 &-3 & 5 &-2 & 1 & 4 & -1& 0 & 1 & 5 & -4
\end{array}$$
We have $4-4-3+5-2+1+4-1+1+5-4=6$, so $133251140379$ has a remainder of $6$ when divided by $13$.

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