document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=752568>Question 752568</A>: <I><SPAN STYLE=\"word-break: break-all;\"> the number of positive integers whose square is a factor of 2000 is out of \r<BR>\n" );
document.write( "\n" );
document.write( "A) 3 <BR>\n" );
document.write( "B) 6<BR>\n" );
document.write( "C)10<BR>\n" );
document.write( "D) 12<BR>\n" );
document.write( "E)20\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you for your help! \r<BR>\n" );
document.write( "\n" );
document.write( "I really don't have a clue how to do this, neither do my friends. And we all don't know WHAT intergers are. Thanks!!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #457862 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=457862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <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( "Positive integers are the counting numbers, the numbers we use to\r\n" );
document.write( "count with: 1,2,3,4,5,6, etc. (ad infinitum)\r\n" );
document.write( "\r\n" );
document.write( "1 is a factor of every positive integer, and the square of 1 is 1,\r\n" );
document.write( "and so 1 fits the requirements.  So one such integer is\r\n" );
document.write( "\r\n" );
document.write( "1. 1\r\n" );
document.write( "\r\n" );
document.write( "To find the others,\r\n" );
document.write( "\r\n" );
document.write( "we break 2000 down into prime factors.\r\n" );
document.write( "\r\n" );
document.write( "We get 2<sup>4</sup>×5<sup>3</sup>\r\n" );
document.write( "\r\n" );
document.write( "The perfect squares we can make of those are \r\n" );
document.write( "\r\n" );
document.write( "2.  2<sup>2</sup>, \r\n" );
document.write( "3.  2<sup>4</sup> = (2<sup>2</sup>)<sup>2</sup> = 4<sup>2</sup> \r\n" );
document.write( "4.  5<sup>2</sup>\r\n" );
document.write( "\r\n" );
document.write( "And when we multiply two squares, we get another square, so we can\r\n" );
document.write( "also get:\r\n" );
document.write( "  \r\n" );
document.write( "5. 2<sup>2</sup>5<sup>2</sup> = (2×5)<sup>2</sup> = 10<sup>2</sup>,\r\n" );
document.write( "6. 4<sup>2</sup>5<sup>2</sup> = (4×5)<sup>2</sup> = 20<sup>2</sup>,\r\n" );
document.write( "\r\n" );
document.write( "So there are 6 such integers:\r\n" );
document.write( "\r\n" );
document.write( "1.  1 is such because 1<sup>2</sup> = 1 = 2000÷2000\r\n" );
document.write( "2.  2 is such because 2<sup>2</sup> = 4 = 2000÷500\r\n" );
document.write( "3.  4 is such because 4<sup>2</sup> = 16 = 2000÷125\r\n" );
document.write( "4.  5 is such because 5<sup>2</sup> = 25 = 2000÷80\r\n" );
document.write( "5.  10 is such because 10<sup>2</sup> = 100 = 2000÷20\r\n" );
document.write( "6.  20 is such because 20<sup>2</sup> = 400 = 2000÷5\r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );