Difference between revisions of "Divisibility rules"
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== Divisibility Rule for 2 and Powers of 2 == | == Divisibility Rule for 2 and Powers of 2 == | ||
− | + | 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>. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisble by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. | |
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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]] |
Revision as of 09:06, 21 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.
Contents
[hide]- 1 Divisibility Rule for 2 and Powers of 2
- 2 Divisibility Rule for 3 and 9
- 3 Divisibility Rule for 5 and Powers of 5
- 4 Divisibility Rule for 7
- 5 Divisibility Rule for 11
- 6 Divisibility Rule for 13
- 7 Divisibility Rule for 17
- 8 Divisibility Rule for 19
- 9 Divisibility Rule for 23
- 10 Divisibility Rule for 29
- 11 Divisibility Rule for 31
- 12 Divisibility Rule for 37
- 13 Divisibility Rule for 41
- 14 Divisibility Rule for 43
- 15 Divisibility Rule for 47
- 16 More general note for primes
- 17 More general note for composites
- 18 Example Problems
- 19 Resources
- 20 See also
Divisibility Rule for 2 and Powers of 2
A number is divisible by if and only if the last digits of the number are divisible by . Thus, in particular, a number is divisible by 2 if and only if its units digit is divisble by 2, i.e. if the number ends in 0, 2, 4, 6 or 8.
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.
Divisibility Rule for 5 and Powers of 5
A number is divisible by if and only if the last digits are divisible by that power of 5.
Divisibility Rule for 7
Rule 1: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 7 if and only if is divisible by 7.
Rule 2: Truncate the last digit of , double that digit, and subtract it from the rest of the number (or vice-versa). is divisible by 7 if and only if the result is divisible by 7.
Divisibility Rule for 11
A number is divisible by 11 if and only if the alternating sum of the digits is divisible by 11.
Divisibility Rule for 13
Rule 1: 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.
Rule 2: Partition into 3 digit numbers from the right (). The alternating sum () is divisible by 13 if and only if is divisible by 13.
Divisibility Rule for 17
Truncate the last digit, multiply it by 5 and subtract from the remaining leading number. THe number is divisible if and only if the result is divisible. The process can be repeated for any number.
Divisibility Rule for 19
Truncate the last digit, multiply it by 2 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers.
Divisibility Rule for 23
Truncate the last digit, multiply it by 7 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :)
Divisibility Rule for 29
Truncate the last digit, multiply it by 3 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O
Divisibility Rule for 31
Truncate the last digit, multiply it by 3 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O
Divisibility Rule for 37
Truncate the last digit, multiply it by 11 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O !
Divisibility Rule for 41
Truncate the last digit, multiply it by 4 and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O
Divisibility Rule for 43
Truncate the last digit, multiply it by 13 and add to the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O !!!
Divisibility Rule for 47
Truncate the last digit, multiply it by 14, and subtract from the remaining leading number. The number is divisible if and only if the result is divisible. This can also be repeated for large numbers. :O !!!!
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 , 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. Also note that these rules exist in two forms: if is replaced by 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 , where the prime factorization of is , if the number is divisible by each of .
Example
Is 55682168544 divisible by 36?
Solution
First, we find the prime factorization of 36 to be . 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.
Example Problems
Resources
Books
- The AoPS Introduction to Number Theory by Mathew Crawford.
- The AoPS Introduction to Number Theory by Sandor Lehoczkyand Richard Rusczyk.
Classes