document.write( "Question 868631: Adding a three-digit number 5Z7 to 386 gives XY3. If XY3 is divisible by 3, then what is the largest possible value of Z? \n" ); document.write( "
Algebra.Com's Answer #523710 by Edwin McCravy(20060)\"\" \"About 
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document.write( " 386\r\n" );
document.write( "+5Z7\r\n" );
document.write( " XY3\r\n" );
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document.write( "The right column adds to 13 so there is 1 to carry to the\r\n" );
document.write( "middle column, so we put a 1 to carry above it:\r\n" );
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document.write( "  1\r\n" );
document.write( " 386\r\n" );
document.write( "+5Z7\r\n" );
document.write( " XY3\r\n" );
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document.write( "Adding the second column:\r\n" );
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document.write( "Either there is or there isn't a 1 to carry to the leftmost column.\r\n" );
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document.write( "We consider the possibility that there is no carry to the leftmost column:\r\n" );
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document.write( "Then Z must be 0 and Y must be 9, and X must be 8:\r\n" );
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document.write( "  1\r\n" );
document.write( " 386\r\n" );
document.write( "+507\r\n" );
document.write( " 893\r\n" );
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document.write( "Trouble is, 893 is not divisible by 3.  So this case is eliminated.\r\n" );
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document.write( "So there must be a carry of 1 to the leftmost column:\r\n" );
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document.write( " 11\r\n" );
document.write( " 386\r\n" );
document.write( "+5Z7\r\n" );
document.write( " XY3\r\n" );
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document.write( "Therefore X can only be 9\r\n" );
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document.write( " 11\r\n" );
document.write( " 386\r\n" );
document.write( "+5Z7\r\n" );
document.write( " 9Y3\r\n" );
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document.write( "The sum XY3 which is 9Y3 must be divisible by 3.  Therefore\r\n" );
document.write( "its sum of digits 9+Y+3 must be a multiple of 3.\r\n" );
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document.write( "Since the sum of the first and third digits of 9Y3 is 9+3=12,\r\n" );
document.write( "which is a multiple of 3, Y must also be divisible by 3.\r\n" );
document.write( "so Y = 0, 3, 6, or 9.\r\n" );
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document.write( "For Z to be as small as possible, Y must be as small as possible:\r\n" );
document.write( "so Y must be 0.\r\n" );
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document.write( " 11\r\n" );
document.write( " 386\r\n" );
document.write( "+5Z7\r\n" );
document.write( " 903\r\n" );
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document.write( "And so Z must be 1:\r\n" );
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document.write( " 11\r\n" );
document.write( " 386\r\n" );
document.write( "+517\r\n" );
document.write( " 903\r\n" );
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document.write( "So the smallest possible value of Z is 1.\r\n" );
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document.write( "Edwin
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