document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006021>Question 1006021</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pronic number is the product of two consecutive positive integers.<BR>\n" );
document.write( "110 is a pronic number as 110 = 11 x 10.<BR>\n" );
document.write( "The sum of two pronic numbers is 240.<BR>\n" );
document.write( "What is the difference between these two numbers? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622370 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=622370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There may be simpler, more efficient ways to solve this, and I would love to hear about them, but here is what I could come up with.<BR>\n" );
document.write( " <BR>\n" );
document.write( "I would look for two positive integers, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > , such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A ALIGN=MIDDLE  ALT=\"A\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B ALIGN=MIDDLE  ALT=\"B\"   DISABLED_style= \"max-width:90%; height:auto;\" > are pronic numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%3EB ALIGN=MIDDLE  ALT=\"A%3EB\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%2BB=240 ALIGN=MIDDLE  ALT=\"A%2BB=240\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240=A%2BB%3E2B%29 ALIGN=MIDDLE  ALT=\"240=A%2BB%3E2B%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=120%3EB%29 ALIGN=MIDDLE  ALT=\"120%3EB%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%2A12=132%3E120 ALIGN=MIDDLE  ALT=\"11%2A12=132%3E120\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B%3C=10%2A11=110 ALIGN=MIDDLE  ALT=\"B%3C=10%2A11=110\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the smallest of the two pronic numbers we are looking for is one of the 10 smallest pronic numbers.<BR>\n" );
document.write( " <BR>\n" );
document.write( "BRUTE FORCE SOLUTION:<BR>\n" );
document.write( "A brute force solution (trying the first ten pronic numbers) looks feasible. <BR>\n" );
document.write( "We could try each of the pronic numbers numbers from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=1%2A2=2 ALIGN=MIDDLE  ALT=\"B=1%2A2=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B=10%2A11=110 ALIGN=MIDDLE  ALT=\"B=10%2A11=110\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and see if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240-B%3C240 ALIGN=MIDDLE  ALT=\"240-B%3C240\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a pronic numbers number .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%2811%2C2%2C%0D%0AB%2C240-B%2C%0D%0A1%2A2=2%2C240-2=238%2C%0D%0A2%2A3=6%2C240-6=234%2C%0D%0A3%2A4=12%2C240-12=228%2C%0D%0A4%2A5=20%2C240-20=220%2C%0D%0A5%2A6=30%2C240-30=210%2C%0D%0A6%2A7=42%2C240-42=198%2C%0D%0A7%2A8=56%2C240-56=184%2C%0D%0A8%2A9=72%2C240-72=168%2C%0D%0A9%2A10=90%2C240-90=150%2C%0D%0A10%2A11=110%2C240-110=130%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "With a list of pronic numbers that would be easy.<BR>\n" );
document.write( " <BR>\n" );
document.write( "With a calculator, I would add the digits <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=25 ALIGN=MIDDLE  ALT=\"25\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the right of each <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240-B ALIGN=MIDDLE  ALT=\"240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" > number,<BR>\n" );
document.write( "and take the square root to see if it is an integer.<BR>\n" );
document.write( "That is a known property of pronic numbers.<BR>\n" );
document.write( "I did not know that, but I took the square roots of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240-B%2B0.25 ALIGN=MIDDLE  ALT=\"240-B%2B0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "until I found that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28210.25%29=14.5 ALIGN=MIDDLE  ALT=\"sqrt%28210.25%29=14.5\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which is the same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2821025%29=145 ALIGN=MIDDLE  ALT=\"sqrt%2821025%29=145\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "I used the fact that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=240-B ALIGN=MIDDLE  ALT=\"240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" > is pronic,<BR>\n" );
document.write( "there is a positive integer <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > such and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29=240-B ALIGN=MIDDLE  ALT=\"n%28n%2B1%29=240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29=240-B ALIGN=MIDDLE  ALT=\"n%28n%2B1%29=240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5-0.5%29%28n%2B0.5%2B0.5%29=240-B ALIGN=MIDDLE  ALT=\"%28n%2B0.5-0.5%29%28n%2B0.5%2B0.5%29=240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5%29%5E2-0.5%5E2=240-B ALIGN=MIDDLE  ALT=\"%28n%2B0.5%29%5E2-0.5%5E2=240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5%29%5E2-0.25=240-B ALIGN=MIDDLE  ALT=\"%28n%2B0.5%29%5E2-0.25=240-B\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5%29%5E2=240-B%2B0.25 ALIGN=MIDDLE  ALT=\"%28n%2B0.5%29%5E2=240-B%2B0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "A MORE ELEGANT SOLUTION:<BR>\n" );
document.write( "Without a calculator,<BR>\n" );
document.write( "I figured I was looking for the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5Eth ALIGN=MIDDLE  ALT=\"n%5Eth\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5Eth ALIGN=MIDDLE  ALT=\"p%5Eth\"   DISABLED_style= \"max-width:90%; height:auto;\" > pronic numbers,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n%2B1%29=%28n%2B0.5%29%5E2-0.25 ALIGN=MIDDLE  ALT=\"n%2A%28n%2B1%29=%28n%2B0.5%29%5E2-0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2A%28p%2B1%29=%28p%2B0.5%29%5E2-0.25 ALIGN=MIDDLE  ALT=\"p%2A%28p%2B1%29=%28p%2B0.5%29%5E2-0.25\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p ALIGN=MIDDLE  ALT=\"p\"   DISABLED_style= \"max-width:90%; height:auto;\" > positive integers, such that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5%29%5E2-0.25%2B%28p%2B0.5%29%5E2-0.25=240 ALIGN=MIDDLE  ALT=\"%28n%2B0.5%29%5E2-0.25%2B%28p%2B0.5%29%5E2-0.25=240\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B0.5%29%5E2%2B%28p%2B0.5%29%5E2=240%2B0.5 ALIGN=MIDDLE  ALT=\"%28n%2B0.5%29%5E2%2B%28p%2B0.5%29%5E2=240%2B0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%28n%2B0.5%29%5E2%2B4%28p%2B0.5%29%5E2=4%2A240%2B4%2A0.5 ALIGN=MIDDLE  ALT=\"4%28n%2B0.5%29%5E2%2B4%28p%2B0.5%29%5E2=4%2A240%2B4%2A0.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282n%2B1%29%5E2%2B%282p%2B1%29%5E2=962 ALIGN=MIDDLE  ALT=\"%282n%2B1%29%5E2%2B%282p%2B1%29%5E2=962\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, I need two odd numbers whose squares add to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=962 ALIGN=MIDDLE  ALT=\"962\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Since odd numbers' squares end in 1, 5, or 9,<BR>\n" );
document.write( "to add up to a sum ending in 2,<BR>\n" );
document.write( "both squares should end in 1.<BR>\n" );
document.write( "If the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%5Eth ALIGN=MIDDLE  ALT=\"p%5Eth\"   DISABLED_style= \"max-width:90%; height:auto;\" > pronic number is the smallest of the two we are looking for,<BR>\n" );
document.write( "and we had initially figured out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%3C=10 ALIGN=MIDDLE  ALT=\"p%3C=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the only possibilities are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%2B1=11 ALIGN=MIDDLE  ALT=\"2p%2B1=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=5 ALIGN=MIDDLE  ALT=\"p=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%2B1=21 ALIGN=MIDDLE  ALT=\"2p%2B1=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p=10 ALIGN=MIDDLE  ALT=\"p=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%2B1=11 ALIGN=MIDDLE  ALT=\"2p%2B1=11\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282p%2B1%29%5E2=121 ALIGN=MIDDLE  ALT=\"%282p%2B1%29%5E2=121\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282n%2B1%29%5E2=962-121=841 ALIGN=MIDDLE  ALT=\"%282n%2B1%29%5E2=962-121=841\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%2B1=sqrt%28841%29=29 ALIGN=MIDDLE  ALT=\"2n%2B1=sqrt%28841%29=29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives us the solution:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n%2B1%29=%28n%2B0.5%29%5E2-0.25=%281%2F4%29%282n%2B1%29%5E2-1%29=%281%2F4%29%28841-1%29=840%2F4=highlight%28210%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2A%28p%2B1%29=%28p%2B0.5%29%5E2-0.25=%281%2F4%29%282p%2B1%29%5E2-1%29=%281%2F4%29%28121-1%29=120%2F4=highlight%2830%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The other choice,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2p%2B1=21 ALIGN=MIDDLE  ALT=\"2p%2B1=21\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282p%2B1%29%5E2=441 ALIGN=MIDDLE  ALT=\"%282p%2B1%29%5E2=441\"   DISABLED_style= \"max-width:90%; height:auto;\" >---><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282n%2B1%29%5E2=962-441=521 ALIGN=MIDDLE  ALT=\"%282n%2B1%29%5E2=962-441=521\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "does not work, because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=521 ALIGN=MIDDLE  ALT=\"521\"   DISABLED_style= \"max-width:90%; height:auto;\" > is not a perfect square. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );