Difference between revisions of "Divisibility"
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Divisibility is the ability of a number to be evenly divided by another number. For example, four divided by two is equal to two, and therefore, four is divisible by two. | Divisibility is the ability of a number to be evenly divided by another number. For example, four divided by two is equal to two, and therefore, four is divisible by two. | ||
| − | == By <math>2^n</math> == | + | == Notation == |
| + | |||
| + | We commonly write <math>n|k</math>. This means that n is a divisor of k. So for the example above, we would write 2|4. | ||
| + | |||
| + | ==Rules for common divisors== | ||
| + | |||
| + | === 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>. | 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>. | ||
| − | == By 3 == | + | === By 3 === |
A number is divisible by 3 if the sum of its digits is divisible by 3. | A number is divisible by 3 if the sum of its digits is divisible by 3. | ||
| − | == By 5^n == | + | ===By <math>5^n</math> === |
A number is divisible by 5^n if the last n digits are divisible by that power of 5. | A number is divisible by 5^n if the last n digits are divisible by that power of 5. | ||
| − | == By 9 == | + | === By 9 === |
A number is divisible by 9 if the sum of its digits is divisible by 9. | A number is divisible by 9 if the sum of its digits is divisible by 9. | ||
| − | == By 7 == | + | === 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.<br> | 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.<br> | ||
<br> | <br> | ||
Rule 2: Truncate the last digit of <math>{n}</math>, and subtract twice that digit from the remaining number. If the result is divisible by 7, then the number is divisible by 7. This process can be repeated for large numbers.<br> | Rule 2: Truncate the last digit of <math>{n}</math>, and subtract twice that digit from the remaining number. If the result is divisible by 7, then the number is divisible by 7. This process can be repeated for large numbers.<br> | ||
| − | == By 11 == | + | === By 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. | ||
| − | == By 13 == | + | === By 13 === |
See rule 1 for divisibility by 7, a number is divisible by 13 if the same specified sum is divisible by 13. | See rule 1 for divisibility by 7, a number is divisible by 13 if the same specified sum is divisible by 13. | ||
Revision as of 17:21, 18 June 2006
Contents
Description
Divisibility is the ability of a number to be evenly divided by another number. For example, four divided by two is equal to two, and therefore, four is divisible by two.
Notation
We commonly write 
. This means that n is a divisor of k. So for the example above, we would write 2|4.
Rules for common divisors
By 
A number is divisible by 
if the last 
digits of the number are divisible by .
By 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
By 
A number is divisible by 5^n if the last n digits are divisible by that power of 5.
By 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
By 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 subtract twice that digit from the remaining number. If the result is divisible by 7, then the number is divisible by 7. This process can be repeated for large numbers.
By 11
A number is divisible by 11 if the alternating sum of the digits is divisible by 11.
By 13
See rule 1 for divisibility by 7, a number is divisible by 13 if the same specified sum is divisible by 13.
