document.write( "Question 1104164: Let a,b,c be integers such that a^2+b^2 = c^2.
\n" ); document.write( "For c divisible by 3, prove that a and b are both divisible by 3 by using congruences. \n" ); document.write( "

Algebra.Com's Answer #718894 by ikleyn(52788) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "For \"c\" be divisible by 3, the necessary and sufficient condition is $\"c%5E2\"$ is divisible by 3.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The Table below contains two inputs:\r\n" );
document.write( "\r\n" );
document.write( "    a) vertical column for \"a mod 3\"  (leftmost column),  and\r\n" );
document.write( "\r\n" );
document.write( "    b) horizontal row  for \"b mod 3\"  (uppermost row).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Next vertical column is \"a^2 mod 3\".\r\n" );
document.write( "\r\n" );
document.write( "Next horizontal row  is \"b^2 mod 3\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The table itself contains the values  \" mod 3\"  in its cells.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "                             0    1    2         <<<---===  b   mod 3\r\n" );
document.write( "\r\n" );
document.write( "   a mod 3    mod 3        0    1    1         <<<---===   mod 3\r\n" );
document.write( "\r\n" );
document.write( "     0          0            0    1    1\r\n" );
document.write( "\r\n" );
document.write( "     1          1            1    2    2\r\n" );
document.write( "\r\n" );
document.write( "     2          1            1    2    2\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From the table, you can see that the sum    is multiple of 3 if and only if both \"a\" and \"b\" are multiples of 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then and only then you have  \"0 mod 3\" in the table.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now the proof is this chain of arguments:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    If \"c\" is multiple of  3,  then    is multiple of 3,   and  since  = ,  \r\n" );
document.write( "\r\n" );
document.write( "    it is possible if and only if  both \"a\"  and  \"b\"  are multiplies of 3.\r\n" );
document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is the shortest way to prove the statement.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );