document.write( "Question 1049663: Ursula has the tiles numbered as 1, 3, 2, 6. How many different ways can she arrange the tiles to form numbers that are divisible by 6? \n" ); document.write( "

Algebra.Com's Answer #665256 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Ursula has the tiles numbered as 1, 3, 2, 6. How many different ways can she arrange the tiles to form numbers that are divisible by 6?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The necessary and sufficient condition for a number to be divisible by 6\r\n" );
document.write( "is to be even and to be divisible by 3.\r\n" );
document.write( "\r\n" );
document.write( "So, the last digit must be 2 or 6. Then and only then the number is even.\r\n" );
document.write( "\r\n" );
document.write( "From the other side, the sum of the digits 1 + 3 + 2 + 6 = 12 is divisible by 3. Hence, any permutation of these digits is the number divisible by 3.\r\n" );
document.write( "\r\n" );
document.write( "Thus any permutation of the digits that ends by 6 or by 2 is multiple of 6.\r\n" );
document.write( "All other permutations are not multiples of 6.\r\n" );
document.write( "\r\n" );
document.write( "Next, there are 3! permutations ending by 2, and \r\n" );
document.write( "      there are 3! permutations ending by 6.\r\n" );
document.write( "\r\n" );
document.write( "In all, there are 3! + 3! = 12 numbers of these digits that are multiple of 6.\r\n" );
document.write( "\r\n" );
document.write( "Answer.  There are 12 numbers of these digits that are multiple of 6.\r\n" );
document.write( "

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

\n" );