document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182798>Question 1182798</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The 4-digit number pqrs has the property that pqrs*4=srqp. If p=2, what is the value of the 3-digit number qrs? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #812888 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=812888\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "In the link from the other tutor, several slightly different algebraic solutions to the problem are shown.<br><BR>\n" );
document.write( "I think the problem is easily solved without algebra, using what we know from early in our mathematics education about multiplication.<br><BR>\n" );
document.write( "We have...<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   pqrs\r\n" );
document.write( "   *  4\r\n" );
document.write( "   ----\r\n" );
document.write( "   srqp</PRE><BR>\n" );
document.write( "and we are told that p is 2.<br><BR>\n" );
document.write( "Note that knowing that p=2 only makes the problem easier to solve; there is in fact only one solution to the problem, and in that solution p is 2.  So let's look at the problem without using the given information that p is 2.  We will quickly determine that by logical analysis.<br><BR>\n" );
document.write( "(1) In the units column of the multiplication, we have s times 4 giving final digit p; that means p is even.<BR>\n" );
document.write( "(2) For the leading digits, we have p times 4 giving us s; and the product is a 4-digit number.<br><BR>\n" );
document.write( "So (1) and (2) tell us p is even, and 4 times p is less than 10.  p can't be 0, so p can only be either 1 or 2.  p \"even\" and \"either 1 or 2\" means p is 2.<br><BR>\n" );
document.write( "So now we have...<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   2qrs\r\n" );
document.write( "   *  4\r\n" );
document.write( "   ----\r\n" );
document.write( "   srq2</PRE><BR>\n" );
document.write( "(3) Now in the units column we have s times 4 giving final digit 2, so s is either 3 or 8.<BR>\n" );
document.write( "(4) And for the leading digits we have 2 times 4 giving final digit s, so s is either 8 or 9.<br><BR>\n" );
document.write( "So (3) and (4) tell us that s must be 8.<br><BR>\n" );
document.write( "So now we have...<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   2qr8\r\n" );
document.write( "   *  4\r\n" );
document.write( "   ----\r\n" );
document.write( "   8rq2</PRE><BR>\n" );
document.write( "(5) For the last two digits, we now have 4 times \"r8\" giving final digit 2.  4 times r is even, and the 4 times 8 gives us a \"carry\" of 3; that means q is odd.<BR>\n" );
document.write( "(6) For the first two digits, we have \"2q\" times 4 giving us \"8r\"; that means q is at most 2.<br><BR>\n" );
document.write( "So q is an odd digit that is at most 2; of course that means q is 1.<br><BR>\n" );
document.write( "So now we have...<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   21r8\r\n" );
document.write( "   *  4\r\n" );
document.write( "   ----\r\n" );
document.write( "   8r12</PRE><BR>\n" );
document.write( "(7) Again for the final two digits we have \"r8\" times 4 giving us tens digit 1.  With the \"carry\" of 3 from the units column, that means r times 4 gives us final digit 8, so r is either 2 or 7.<BR>\n" );
document.write( "(8) And for the leading two digits, \"21\" times 4 gives us leading digits \"8r\", so r can be 4, 5, 6, or 7.<br><BR>\n" );
document.write( "And that tells us r must be 7.<br><BR>\n" );
document.write( "So now we have the unique final answer for pqrs times 4 being equal to srqp:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "   2178\r\n" );
document.write( "   *  4\r\n" );
document.write( "   ----\r\n" );
document.write( "   8712</PRE><BR>\n" );
document.write( "Solved... without algebra.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );