document.write( "Question 520127: Prove that every positive integer a, written in the base 10, a^5 and a have the same last digit. \n" ); document.write( "

Algebra.Com's Answer #345932 by solver91311(24713) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The last digit of an integer is the remainder when the number is divided by 10. Any two natural numbers

share the same last digit exactly when

(Note:

denotes \"divides\" and

denotes \"does not divide\")\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "First prove

: If

is even, then

is even and the difference of two even numbers is even. If

is odd, then

is odd, and the difference of two odd numbers is even. Thus

.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Next prove

. Fermat's Little Theorem: If

is prime, for any integer

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Hence

for some integers

. That means

. Since

, so

since 2 is prime. Then

where

is an integer, and then

. \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Therefore

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "John
\n" ); document.write( "

\n" ); document.write( "My calculator said it, I believe it, that settles it
\n" ); document.write( "

$\"The$

\n" ); document.write( " \n" ); document.write( "

\n" );