document.write( "Question 1199610: Each letter in this multiplication problem stands for a different digit and every letter always stands for the same digit wherever it appears.What is the value of TROUT?
\n" ); document.write( " ROOT
\n" ); document.write( "x U
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\n" ); document.write( "TROUT \n" ); document.write( "
Algebra.Com's Answer #833559 by greenestamps(13200)\"\" \"About 
You can put this solution on YOUR website!


\n" ); document.write( "You won't get any benefit from this if we solve it for you....

\n" ); document.write( "So I will demonstrate HOW to solve it and let you do the work to actually find the answer.
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document.write( "   R O O T\r\n" );
document.write( "  x      U\r\n" );
document.write( " ---------\r\n" );
document.write( " T R O U T\r\n" );
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document.write( "For the discussion, I am replacing letter O with a different letter, to avoid confusion with digit 0.\r\n" );
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document.write( "   R A A T\r\n" );
document.write( "  x      U\r\n" );
document.write( " ---------\r\n" );
document.write( " T R A U T\r\n" );
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document.write( "In the units column, T times U gives final digit T; and we know U can't be 1.  So find the possible combinations of values for (T,U) that satisfy all the conditions:\r\n" );
document.write( "U not = 1\r\n" );
document.write( "T, U different\r\n" );
document.write( "T times U gives last digit T\r\n" );
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document.write( "if U=2, there is no possible value for T\r\n" );
document.write( "if U=3, then T can only be 5: (5,3)\r\n" );
document.write( "if U=4, there is no possible value for T\r\n" );
document.write( "if U=5, there is no possible value for T\r\n" );
document.write( "if U=6, then T can be any of 2, 4, or 8: (2,6)  (4,6)  (8,6)\r\n" );
document.write( "if U=7, then T can only be 5: (5,7)\r\n" );
document.write( "if U=8, there is no possible value for T\r\n" );
document.write( "if U=9, then T can only be 5: (5,9)\r\n" );
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document.write( "The possible combinations for (T,U) are\r\n" );
document.write( "(5,3)\r\n" );
document.write( "(5,7)\r\n" );
document.write( "(5,9)\r\n" );
document.write( "(2,6)\r\n" );
document.write( "(4,6)\r\n" );
document.write( "(8,6)\r\n" );
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document.write( "Now try each combination and find the one that does not lead to a contradiction.\r\n" );
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document.write( "(T,U) = (2,6)\r\n" );
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document.write( "   R A A 2\r\n" );
document.write( "  x      6\r\n" );
document.write( " ---------\r\n" );
document.write( " 2 R A 6 2\r\n" );
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document.write( "Here, 2 times 6 in the units column is 12, giving us 2 in the units column of the product and a carry of 1.  But then in the tens column we have 6 times A plus a carry of 1 giving us digit 6 in the product.  But for any digit A, 6 times A is even, so 6 times A plus 1 is odd.\r\n" );
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document.write( "So (T,U) = (2,6) is not a solution.\r\n" );
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document.write( "(T,U) = (5,7)\r\n" );
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document.write( "   R A A 5\r\n" );
document.write( "  x      7\r\n" );
document.write( " ---------\r\n" );
document.write( " 5 R A 7 5\r\n" );
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document.write( "Here, 5 times 7 in the units column is 35, so we have 5 in the units column of the product, with a carry of 3.\r\n" );
document.write( "So in the tens column, we have 7 times A plus a carry of 3 giving digit 7 in the product.  The only possible value for A that gives us that is 2, which gives us a 7 in the tens column of the product and a carry of 1.  So\r\n" );
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document.write( "     1 3\r\n" );
document.write( "   R 2 2 5\r\n" );
document.write( "  x      7\r\n" );
document.write( " ---------\r\n" );
document.write( " 5 R 2 7 5\r\n" );
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document.write( "But this gives us, in the hundreds column, 7 times 2 plus 1 giving last digit 5; but we know it has to be 2.\r\n" );
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document.write( "So (T,U) = (5,7) is not a solution.
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\n" ); document.write( "\n" ); document.write( "Try the other possibilities for (T,U) until you find the solution....

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