document.write( "Question 1209560: Hi,\r
\n" ); document.write( "\n" ); document.write( " abc
\n" ); document.write( "+ de
\n" ); document.write( " be
\n" ); document.write( " =bdc, what is the solution \n" ); document.write( "

Algebra.Com's Answer #849526 by greenestamps(13200) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "I will assume the cryptarithm is this:

\r\n" );
document.write( "\r\n" );
document.write( "  ABC\r\n" );
document.write( "   DE\r\n" );
document.write( " + BE\r\n" );
document.write( "  ---\r\n" );
document.write( "  BDC

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\n" ); document.write( "\n" ); document.write( "This is not a very well-made cryptarithm; there are many solutions.

\n" ); document.write( "(1) In the units column, C+E+E yields final digit C. That means E is either 0 or 5.

\n" ); document.write( "(2) In the tens column, B+D+B, plus possibly a carry of 1, yields digit D in the sum, so B is also either 0 or 5.

\n" ); document.write( "(3) Since in the tens column B+D+B yields digit D in the sum, there can not be a carry of 1 from the units column.

\n" ); document.write( "(4) From (1), (2), and (3), E must be 0 and B must be 5.

\n" ); document.write( "(5) In the hundreds column, A plus a carry of 1 yields leading digit B (=5) in the sum, so A is 4.

\n" ); document.write( "So we know the addition looks like this:

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document.write( "\r\n" );
document.write( "  45C\r\n" );
document.write( "   D0\r\n" );
document.write( " + 50\r\n" );
document.write( "  ---\r\n" );
document.write( "  5DC

\n" ); document.write( "There are no further restrictions, so C and D can be any digits. The three digits 0, 4, and 5 have been used; assuming the letters represent different digits, there are 7 choices remaining for C and then 6 choices remaining for D, making a total of 7*6 = 42 solutions.

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