document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=420125>Question 420125</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For how many integers n is n/(20-n) the square of an integer? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #293682 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=293682\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We could assign k to be an integer such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2F%2820-n%29+=+k%5E2 ALIGN=MIDDLE  ALT=\"n%2F%2820-n%29+=+k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Solve for n and you will get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+k%5E2%2820-n%29+=+20k%5E2+-+nk%5E2 ALIGN=MIDDLE  ALT=\"n+=+k%5E2%2820-n%29+=+20k%5E2+-+nk%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%281%2Bk%5E2%29+=+20k%5E2 ALIGN=MIDDLE  ALT=\"n%281%2Bk%5E2%29+=+20k%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+20k%5E2%2F%281%2Bk%5E2%29+=+20+-+%2820%2F%281%2Bk%5E2%29%29 ALIGN=MIDDLE  ALT=\"n+=+20k%5E2%2F%281%2Bk%5E2%29+=+20+-+%2820%2F%281%2Bk%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Here, the positive factors of 20 are 1,2,5,10,20, so k^2 can equal 0,1,4,9, or 19. This implies that k = {0,1,2,3} and n = 0, 10, 16, or 18.\r<BR>\n" );
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document.write( "Another way just as effective is to brute-force the problem. Most math teachers recommend against this, but I will sometimes brute-force a problem if I know for sure the method will work, if I know I'm counting all possible cases, and if the method does not take too much time. Here, we know that n can only be between 0 and 20; otherwise, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2F%2820-n%29 ALIGN=MIDDLE  ALT=\"n%2F%2820-n%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be negative. </SPAN>\n" );
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