document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=783031>Question 783031</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If the product of the integers a, b, and c is 1, then what is the smallest possible value of (a+1)(b+1)(c+1)?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #496513 by </B><A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><B>tommyt3rd(5050)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tommyt3rd><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=tommyt3rd><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=496513\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Expanding and simplifying...<BR>\n" );
document.write( "(a+1)(b+1)(c+1)=<BR>\n" );
document.write( "(a+1)(bc+b+c+1)=<BR>\n" );
document.write( "abc+ab+ac+bc+a+b+c+1=<BR>\n" );
document.write( "ab+bc+ac+a+b+c+2\r<BR>\n" );
document.write( "\n" );
document.write( "Since their product is 1, the only possibilities are a,b,c={-1,1}  (why?)<BR>\n" );
document.write( "Furthermore, there must be an odd number of positives. To find the minimum we can ignore the case a=b=c=1 - that would be the maximum.\r<BR>\n" );
document.write( "\n" );
document.write( "Case1<BR>\n" );
document.write( "if say a=b=1, c=-1, then\r<BR>\n" );
document.write( "\n" );
document.write( "ab+bc+ac+a+b+c+2=<BR>\n" );
document.write( "1-1-1+1+1-1+2=<BR>\n" );
document.write( "2\r<BR>\n" );
document.write( "\n" );
document.write( "Case2<BR>\n" );
document.write( "if say c=1, a=b=-1, then\r<BR>\n" );
document.write( "\n" );
document.write( "ab+bc+ac+a+b+c+2=<BR>\n" );
document.write( "1-1-1-1-1+1+2=<BR>\n" );
document.write( "0\r<BR>\n" );
document.write( "\n" );
document.write( "Minimum value is 0 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );