Difference between revisions of "Divisibility rules"
m (→Divisibility Rule for 13: Proof needs quick finish) |
|||
Line 5: | Line 5: | ||
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>. | 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>. | ||
+ | [[Divisibility rules/Rule for 2 and powers of 2 proof | Proof]] | ||
== Divisibility Rule for 3 and 9== | == Divisibility Rule for 3 and 9== | ||
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. | 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. | ||
+ | |||
+ | [[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 == | ||
− | A number is divisible by <math>5^n</math> if the last | + | A number is divisible by <math>5^n</math> 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 Rule for 11 == | == Divisibility Rule for 11 == | ||
A number is divisible by 11 if the alternating sum of the digits is divisible by 11. | A number is divisible by 11 if the alternating sum of the digits is divisible by 11. | ||
+ | [[Divisibility rules/Rule for 11 proof | Proof]] | ||
===Proof=== | ===Proof=== | ||
Line 55: | Line 30: | ||
== Divisibility Rule for 7 == | == Divisibility Rule for 7 == | ||
− | 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. | + | 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. |
− | + | ||
+ | [[Divisibility rules/Rule 1 for 7 proof | Proof]] | ||
+ | |||
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> | 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> | ||
+ | |||
+ | [[Divisibility rules/Rule 2 for 7 proof | Proof]] | ||
===Proof for Rule 2:=== | ===Proof for Rule 2:=== | ||
Line 65: | Line 44: | ||
== Divisibility Rule for 13 == | == Divisibility Rule for 13 == | ||
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. | 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. | ||
+ | |||
+ | [[Divisibility rules/Rule for 13 proof | Proof]] | ||
===Proof=== | ===Proof=== |
Revision as of 21:51, 15 August 2006
These divisibility rules help determine when integers are divisible by particular other integers.
Contents
[hide]Divisibility Rule for 2 and Powers of 2
A number is divisible by if the last digits of the number are divisible by .
Divisibility Rule for 3 and 9
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.
Divisibility Rule for 5 and Powers of 5
A number is divisible by if the last digits are divisible by that power of 5.
Divisibility Rule for 11
A number is divisible by 11 if the alternating sum of the digits is divisible by 11.
Proof
. Since the number is in base 10, as we assume, the digits would be of powers of 10, or in mod 11, powers of -1. The number 3806 thus becomes:
.
Divisibility Rule for 7
Rule 1: Partition into 3 digit numbers from the right (). If the alternating sum () is divisible by 7, then the number is divisible by 7.
Rule 2: Truncate the last digit of , 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.
Proof for Rule 2:
The divisibility rule would be , where , where is the nth digit from the right (NOT the left) and we have and since 2 is relatively prime to 7, . Then yet again , and this is equivalent to our original number.
Divisibility Rule for 13
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.
Proof
Let be a positive integer with units digit , tens digit and so on. Then is the result of truncating the last digit from . Note that . Now if and only if , from which the rule follows ... (someone add the last 2 lines of pf.)
More general note
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 , there exists some number such that an integer is divisible by if and only if truncating the last digit, multiplying it by and subtracting it from the remaining number gives us a result divisible by . Divisibility rule 2 for 7 says that for , . The divisibility rule for 11 is equivalent to choosing . The divisibility rule for 3 is equivalent to choosing . These rules can also be found under the appropriate conditions in number bases other than 10.
Example Problems
Resources
Books
- The AoPS Introduction to Number Theory by Mathew Crawford.
Classes