Difference between revisions of "Divisibility rules/Rule for 2 and powers of 2 proof"
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| <math>N</math> || <math>= 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10 a_1 + a_0</math> | | <math>N</math> || <math>= 10^k a_k + 10^{k-1} a_{k-1} + \cdots + 10 a_1 + a_0</math> | ||
|- | |- | ||
− | | || <math>\displaystyle \equiv 10^{n-1} a_{n-1} + 10^{n-2} a_{n-2} + \cdots + a_1 + a_0 | + | | || <math>\displaystyle \equiv 10^{n-1} a_{n-1} + 10^{n-2} a_{n-2} + \cdots + a_1 + a_0 </math> |
|- | |- | ||
− | | || <math> | + | | || <math>\equiv a_{n-1}a_{n-2}\cdots a_1a_0 \pmod{2^n} </math> |
|} | |} | ||
+ | |||
+ | Thus, if the last <math>n</math> digits of <math>N</math> are divisible by <math>2^n</math> then <math>N</math> is divisible by <math>2^n</math>. | ||
+ | |||
+ | == See also == | ||
+ | * [[Divisibility rules | Back to divisibility rules]] | ||
== See also == | == See also == | ||
* [[Divisibility rules | Back to divisibility rules]] | * [[Divisibility rules | Back to divisibility rules]] |
Revision as of 23:21, 16 August 2006
A number is divisible by if the last digits of the number are divisible by .
Proof
An understanding of basic modular arithmetic is necessary for this proof.
Let be the base-ten expression for , where the are digits.
Thus
Taking mod gives
Thus, if the last digits of are divisible by then is divisible by .