Divisibility By 7 Revisited

One of the most-viewed posts on this blog describes a rule for checking whether a number is divisible by seven.

This post is about another, simpler way to do it.

Take a long number, say

960,937,563,483

One way to check if it’s divisible by 7 is to subtract multiples of seven until you get down to something small. For example, 910,000,000,000 is a multiple of seven because 91 is. So subtract this number from the original to get 50,937,563,483. Now subtract 49,000,000,000 to get 1,937,563,483, etc.

This procedure is fine for small numbers, but you’ll only eliminate one or two digits per subtraction. Here’s a method useful for very long numbers.

First, turn all the 7‘s into 0‘s, all the 8‘s into 1‘s, and all the 9‘s into 2‘s. This is just a simple case of the rule above – subtracting some multiples of 7. The original number becomes

260,230,563,413

(If you want, you can take this step further by turning 6 into -1, etc. I won’t do that here.)

Now break apart each group of three digits separated by parentheses, starting from the right. Put a negative sign on every other group of three digits, then add them all up.

\begin{array}{cc} {} & +413 \\ {} & -563 \\ {} & +230 \\ + & \underline{-260} \\ {} & -180 \end{array}

This number is divisible by 7 if and only if the original is.

We need to check -180 for divisibility by 7. Here, adding and subtracting multiples of 7 is easy. For example, add 210 to get 30. Now subtract 28 to get 2. The remainder when you divide 960,937,563,483 by 7 is 2.

If you want to use this rule, but the number of digits isn’t a multiple of three, you can simple add some zeros on front. For example,

45,338,017

turns into

045,331,010

and we get

\begin{array}{cc} {} & +010 \\ {} & -331 \\+ & \underline{+045} \\ {} & -276 \end{array}

-276 + 280 = 4, so the remainder when you divide 45,338,017 by 7 is 4.

This rule works due to a convenient pattern in the remainders of the powers of ten.

If we start with n = 0, the remainder when you divide powers of 10 by 7 is

1, 3, 2, -1, -3, -2, 1, 3, 2, -1, -3, -2,  \ldots

Each group of three digits, after alternating groups are multiplied by -1, has the same rule for divisibility by 7. For example, the remainder when 124,000,000,000 is divided by seven is the same as the remainder for -124. So we just take those groups of three and add them, simplifying the task greatly.

One Response to “Divisibility By 7 Revisited”

  1. Multiples Rule for 7 « Arcsecond Says:

    […] Update: check out a new rule for divisibility by seven. […]

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