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Difference between revisions of "Divisibility rules"

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These '''divisibility rules''' help determine when [[positive integer]]s are [[divisibility | divisible]] by particular other [[integer]]s.
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These '''divisibility rules''' help determine when [[positive integer]]s are [[divisibility | divisible]] by particular other [[integer]]s.  All of these rules apply for [[Base number| base-10]] ''only'' -- other bases have their own, different versions of these rules.
  
  
 
== Divisibility Rule for 2 and Powers of 2 ==
 
== Divisibility Rule for 2 and Powers of 2 ==
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A number is divisible by <math>2^n</math> if the last <math>{n}</math> digits of the number are divisible by <math>2^n</math>.
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A number is divisible by <math>2^n</math> if and only if the last <math>{n}</math> digits of the number are divisible by <math>2^n</math>.
  
 
[[Divisibility rules/Rule for 2 and powers of 2 proof | Proof]]
 
[[Divisibility rules/Rule for 2 and powers of 2 proof | Proof]]
  
 
== Divisibility Rule for 3 and 9==
 
== Divisibility Rule for 3 and 9==
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A number is divisible by 3 or 9 if the sum of its digits is divisible by 3 or 9, respectively.  Note that this does ''not'' work for higher powers of 3.  For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.
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A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively.  Note that this does ''not'' work for higher powers of 3.  For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.
  
 
[[Divisibility rules/Rule for 3 and 9 proof | Proof]]
 
[[Divisibility rules/Rule for 3 and 9 proof | Proof]]
  
 
== Divisibility Rule for 5 and Powers of 5 ==
 
== Divisibility Rule for 5 and Powers of 5 ==
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A number is divisible by <math>5^n</math> if the last <math>n</math> digits are divisible by that power of 5.
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A number is divisible by <math>5^n</math> if and only if the last <math>n</math> digits are divisible by that power of 5.
  
 
[[Divisibility rules/Rule for 5 and powers of 5 proof | Proof]]
 
[[Divisibility rules/Rule for 5 and powers of 5 proof | Proof]]
  
 
== Divisibility Rule for 7 ==
 
== Divisibility Rule for 7 ==
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Rule 1:  Partition <math>n</math> into 3 digit numbers from the right (<math>d_3d_2d_1,d_6d_5d_4,\dots</math>).  If the alternating sum (<math>d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots</math>) is divisible by 7, then the number is divisible by 7.
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Rule 1:  Partition <math>N</math> into 3 digit numbers from the right (<math>d_3d_2d_1,d_6d_5d_4,\dots</math>).  The alternating sum (<math>d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots</math>) is divisible by 7 if and only if <math>N</math> is divisible by 7.
  
 
[[Divisibility rules/Rule 1 for 7 proof | Proof]]
 
[[Divisibility rules/Rule 1 for 7 proof | Proof]]
  
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Rule 2:  Truncate the last digit of <math>{n}</math>, and double that digit, subtracting the rest of the number from the doubled last digit.  If the absolute value of the result is a multiple of 7, then the number itself is. <br>
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Rule 2:  Truncate the last digit of <math>N</math>, double that digit, and subtract it from the rest of the number (or vice-versa).  <math>N</math> is divisible by 7 if and only if the result is divisible by 7.
  
 
[[Divisibility rules/Rule 2 for 7 proof | Proof]]
 
[[Divisibility rules/Rule 2 for 7 proof | Proof]]
  
 
== Divisibility Rule for 11 ==
 
== Divisibility Rule for 11 ==
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A number is divisible by 11 if the alternating sum of the digits is divisible by 11.
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A number is divisible by 11 if and only if the [[alternating sum]] of the digits is divisible by 11.
  
 
[[Divisibility rules/Rule for 11 proof | Proof]]
 
[[Divisibility rules/Rule for 11 proof | Proof]]
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== Divisibility Rule for 13 ==
 
== Divisibility Rule for 13 ==
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Multiply the last digit by 4 and add it to the rest of the number.  This process can be repeated for large numbers, as with the second divisibility rule for 7.   
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Truncate the last digit, multiply it by 4 and add it to the rest of the number.  The result is divisible by 13 if and only if the original number was divisble by 13.  This process can be repeated for large numbers, as with the second divisibility rule for 7.   
  
 
[[Divisibility rules/Rule for 13 proof | Proof]]
 
[[Divisibility rules/Rule for 13 proof | Proof]]
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==More general note for primes ==
 
==More general note for primes ==
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For every [[prime number]] other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7.  For a general prime <math>p</math>, there exists some number <math>q</math> such that an integer is divisible by <math>p</math> if and only if truncating the last digit, multiplying it by <math>q</math> and subtracting it from the remaining number gives us a result divisible by <math>p</math>.  Divisibility rule 2 for 7 says that for <math>p = 7</math>, <math>q = 2</math>.  The divisibility rule for 11 is equivalent to choosing <math>q = 1</math>.  The divisibility rule for 3 is equivalent to choosing <math>q = -1</math>.  These rules can also be found under the appropriate conditions in [[number base]]s other than 10.
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For every [[prime number]] other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7.  For a general prime <math>p</math>, there exists some number <math>q</math> such that an integer is divisible by <math>p</math> if and only if truncating the last digit, multiplying it by <math>q</math> and subtracting it from the remaining number gives us a result divisible by <math>p</math>.  Divisibility rule 2 for 7 says that for <math>p = 7</math>, <math>q = 2</math>.  The divisibility rule for 11 is equivalent to choosing <math>q = 1</math>.  The divisibility rule for 3 is equivalent to choosing <math>q = -1</math>.  These rules can also be found under the appropriate conditions in [[number base]]s other than 10.  Also note that these rules exist in two forms: if <math>q</math> is replaced by <math>p - q</math> then subtraction may be replaced with addition.  We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding.
  
 
== More general note for composites ==
 
== More general note for composites ==

Revision as of 09:05, 17 August 2006

These divisibility rules help determine when positive integers are divisible by particular other integers. All of these rules apply for base-10 only -- other bases have their own, different versions of these rules.


Divisibility Rule for 2 and Powers of 2

A number is divisible by $2^n$ if and only if the last ${n}$ digits of the number are divisible by $2^n$.

Proof

Divisibility Rule for 3 and 9

A number is divisible by 3 or 9 if and only if the sum of its digits is divisible by 3 or 9, respectively. Note that this does not work for higher powers of 3. For instance, the sum of the digits of 1899 is divisible by 27, but 1899 is not itself divisible by 27.

Proof

Divisibility Rule for 5 and Powers of 5

A number is divisible by $5^n$ if and only if the last $n$ digits are divisible by that power of 5.

Proof

Divisibility Rule for 7

Rule 1: Partition $N$ into 3 digit numbers from the right ($d_3d_2d_1,d_6d_5d_4,\dots$). The alternating sum ($d_3d_2d_1 - d_6d_5d_4 + d_9d_8d_7 - \dots$) is divisible by 7 if and only if $N$ is divisible by 7.

Proof

Rule 2: Truncate the last digit of $N$, double that digit, and subtract it from the rest of the number (or vice-versa). $N$ is divisible by 7 if and only if the result is divisible by 7.

Proof

Divisibility Rule for 11

A number is divisible by 11 if and only if the alternating sum of the digits is divisible by 11.

Proof


Divisibility Rule for 13

Truncate the last digit, multiply it by 4 and add it to the rest of the number. The result is divisible by 13 if and only if the original number was divisble by 13. This process can be repeated for large numbers, as with the second divisibility rule for 7.

Proof


More general note for primes

For every prime number other than 2 and 5, there exists a rule similar to rule 2 for divisibility by 7. For a general prime $p$, there exists some number $q$ such that an integer is divisible by $p$ if and only if truncating the last digit, multiplying it by $q$ and subtracting it from the remaining number gives us a result divisible by $p$. Divisibility rule 2 for 7 says that for $p = 7$, $q = 2$. The divisibility rule for 11 is equivalent to choosing $q = 1$. The divisibility rule for 3 is equivalent to choosing $q = -1$. These rules can also be found under the appropriate conditions in number bases other than 10. Also note that these rules exist in two forms: if $q$ is replaced by $p - q$ then subtraction may be replaced with addition. We see one instance of this in the divisibility rule for 13: we could multiply by 9 and subtract rather than multiplying by 4 and adding.

More general note for composites

A number is divisible by $N$, where the prime factorization of $N$ is $p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}$, if the number is divisible by each of $p_1^{e_1}, p_2^{e_2},\ldots, p_n^{e_n}$.

Example

Is 55682168544 divisible by 36?

Solution

First, we find the prime factorization of 36 to be $2^2\cdot 3^2$. Thus we must check for divisibility by 4 and 9 to see if it's divisible by 36.

Since the last two digits, 44, of the number is divisible by 4, so is the entire number.

To check for divisibility by 9, we look to see if the sum of the digits is divisible by 9. The sum of the digits is 54 which is divisible by 9.

Thus, the number is divisible by both 4 and 9 and must be divisible by 36.

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