document.write( "Question 376068: for all numbers x and y, let the operation _ be defined by x _ y = xy - y. If a and b are positive integers, which of the following can be equal to zero?\r
\n" ); document.write( "\n" ); document.write( "** the underscore or _ stands for a missing operation\r
\n" ); document.write( "\n" ); document.write( "1.) a _ b
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\n" ); document.write( "\n" ); document.write( "a.) one only
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Algebra.Com's Answer #267500 by Edwin McCravy(20056)\"\" \"About 
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document.write( "x _ y = xy - y\r\n" );
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document.write( "1.) a _ b\r\n" );
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document.write( "That means ab - b. \r\n" );
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document.write( "Let's set\r\n" );
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document.write( "ab - b = 0\r\n" );
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document.write( "b(a - 1) = 0\r\n" );
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document.write( "Using the zero factor property,\r\n" );
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document.write( "b = 0 or  a - 1 = 0\r\n" );
document.write( "              a = 1\r\n" );
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document.write( "Since b is a positive integer we can rule out b = 0.\r\n" );
document.write( "However if a = 1  then ab - b =0 when b is any positive integer. \r\n" );
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document.write( "So 1.) can be zero.\r\n" );
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document.write( "-------------------------\r\n" );
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document.write( "2.) (a+b)_b\r\n" );
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document.write( "That means (a+b)b - b.  \r\n" );
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document.write( "Let's set\r\n" );
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document.write( "(a+b)b - b = 0\r\n" );
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document.write( "Factor out b:\r\n" );
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document.write( "b[(a+b)-1] = 0\r\n" );
document.write( "b[a+b-1] = 0\r\n" );
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document.write( "Using the zero-factor property again:\r\n" );
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document.write( "b = 0;  a+b-1 = 0\r\n" );
document.write( "            a = 1-b\r\n" );
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document.write( "We must rule out b = 0 since b must be a positive integer.\r\n" );
document.write( "We also must rule out a = 1-b since a positive integer b subtracted\r\n" );
document.write( "from 1 canNOT be a positive integer.\r\n" );
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document.write( "So (2) canNOT be zero.\r\n" );
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document.write( "----------------------------\r\n" );
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document.write( "3. a_(a+b)\r\n" );
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document.write( "That means a(a+b) - (a+b).  \r\n" );
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document.write( "Let's set:\r\n" );
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document.write( "a(a+b) - (a+b) = 0\r\n" );
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document.write( "Factor out (a+b)\r\n" );
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document.write( "(a+b)(a-1) = 0\r\n" );
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document.write( "Using the zero-factor property again:\r\n" );
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document.write( "a+b = 0 ;  a-1 = 0\r\n" );
document.write( "  a = -b;    a = 1\r\n" );
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document.write( "We have to rule out a = -b because a and b are both positive integers,\r\n" );
document.write( "but if a = 1, then\r\n" );
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document.write( "(1+b)(1-1) = (1+b)0 = 0, regardless of what positive integer b equals.\r\n" );
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document.write( "So 3.) can be zero\r\n" );
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document.write( "-----------------------\r\n" );
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document.write( "So only\r\n" );
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document.write( "2.) (a+b)_b\r\n" );
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document.write( "cannot be zero.  The other two can be zero.\r\n" );
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document.write( "Therefore the correct choice is\r\n" );
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document.write( "e.) one and three\r\n" );
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document.write( "Edwin
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