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We know that a number is divisible by $11$ if the difference of sum of odd placed digits and the sum of even placed digits is divisible by $11$.

For example, the number $1023$ is divisible by $11$:

$1+2=3$

$0+3=3$

$3-3=0$

$0$ is divisible by $11$. so $1023$ is divisible by $11$.

Also we can say $1-0+2-3=0$.

On the other hand there are unique rules for divisibility by odd prime numbers. For example, to find if a number is divisible by $7$, we should multiply the unit digit by $2$, then substract the remaining number from that. For example, for $343$ we have $34-6=28$ and $28$ is divisible by $7$.

My question: Is there such a rule for divisibility by $11$ as well?

Bill Dubuque
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  • The Wikipedia page on all well known divisibility rules can be helpful. In the case of 11, the page mentions 6 such rules. There exists a rule that is similar to that of 7 in which you subtract the units digit from the number formed by the remaining digits and check if it’s divisible by 11. – Sahaj Aug 05 '23 at 16:01
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    What do you mean by "unique"? For every number you can come up with multiple divisibility rules. In which way is that particular rule for $7$ unique, that that particular one for $11$ isn't? – NDB Aug 05 '23 at 16:05
  • Dubuque always making duplicates that make no sense. – NDB Aug 05 '23 at 16:14
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    The "other rule" is equivalent, viz. $11\mid 10b+a \iff 11\mid b-a,,$ which when iterated yields the common test using the alternating digit sum - see the linked dupes. – Bill Dubuque Aug 05 '23 at 16:19

1 Answers1

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There is one for each prime number different from 2 and 5. Suppose $10a + b$ is your number, where $b$ is just the last digit, and $p$ is your prime. You have that $$ p|10a + b \iff p| a + (10)^{-1}b $$ In other words, you just have to find the inverse of $10$ (i.e. a number $-p<k<p$ such that $10 k$ is a multiple of $p$ plus 1) modulo $p$.

For $p=7$ we have $k=5$ since $5*10=50 = 49+1$, but also $k=-2$ since $-2*10 = -20 = -21 +1$.

For $p=11$ we have $k=-1$ since $-1*10 = -10 = -11 +1$. In other words, you just need to subtract the unit digit from the rest of the number.

Exodd
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