document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161905>Question 1161905</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose your textbook is opened at random and the product of the facing page numbers is 71556. To what page is the textbook opened? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785543 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=785543\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "An algebraic approach\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = unknown page number<BR>\n" );
document.write( "x+1 = page number after page x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x*(x+1) = x^2+x = product of page numbers<BR>\n" );
document.write( "x^2+x = 71556<BR>\n" );
document.write( "x^2+x-71556 = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the quadratic formula (with a = 1, b = 1, c = -71556) to find the two solutions to be x = 267 and x = -268. Ignore the negative x value, because we can't have negative page numbers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the only practical solution is x = 267. The pages we've opened the book to are 267 and 268.<BR>\n" );
document.write( "We see that 267*268 = 71556 which confirms the answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Factoring or graphing is an alternative to solve x^2+x-71556 = 0; though both aren't really feasible by hand. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "-------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A non-algebraic approach is to look at the prime factorization of 71556<BR>\n" );
document.write( "71556 = 2*2*3*67*89\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then group those factors into two groups, call them left and right group<BR>\n" );
document.write( "One such arrangement is<BR>\n" );
document.write( "left group = {2,2,3}<BR>\n" );
document.write( "right group = {67,89}<BR>\n" );
document.write( "The ordering of any particular single grouping does not matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then multiply out each of the two groups<BR>\n" );
document.write( "left group's product = 2*2*3 = 12<BR>\n" );
document.write( "right group's product = 67*89 = 5963<BR>\n" );
document.write( "the results of each group will then be the two page numbers; however, notice that 12 and 5963 are not adjacent pages. Also, it is unlikely a textbook has 5963 pages or more.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then another arrangement is<BR>\n" );
document.write( "left group = {2,67,3}<BR>\n" );
document.write( "right group = {2*89}<BR>\n" );
document.write( "left group's product = first page number = 2*67*3 = 402<BR>\n" );
document.write( "right group's product = second page number = 2*89 = 178<BR>\n" );
document.write( "We again get non-adjacent page numbers. At least the values are reasonable for any textbook.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the idea is to guess and check. It turns out after a bit of trial and error you should find that<BR>\n" );
document.write( "left group = 3*89 = 267<BR>\n" );
document.write( "right group = 2*2*67 = 268<BR>\n" );
document.write( "I used the previous section to help get to this quickly. If you didn't have that as a guide, then it would take longer. Though it's still a somewhat feasible route.<BR>\n" );
document.write( "The recommendation I have is to make sure 67 and 89 are not in the same group. Otherwise their product will multiply to 67*89 = 5963 (or larger if other factors are involved).<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );