document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1133242>Question 1133242</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the smallest positive integer which is 2 times the square of some<BR>\n" );
document.write( "positive integer and also 5 times the fifth power of some other positive integer?<BR>\n" );
document.write( "  </SPAN>\n" );
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document.write( "We are looking for the smallest positive integer that is both 2 times the square of some positive integer and 5 times the 5th power of another positive integer:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+=+5y%5E5 ALIGN=MIDDLE  ALT=\"2x%5E2+=+5y%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There is a single known factor of 2 on the left and a single known factor of 5 on the right.  That means both x and y must both contain factors of only 2 and/or 5.  So<br><BR>\n" );
document.write( "let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+%282%5Ea%29%285%5Eb%29 ALIGN=MIDDLE  ALT=\"x%5E2+=+%282%5Ea%29%285%5Eb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where a and b are even; and<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E5+=+%282%5Ec%29%285%5Ed%29 ALIGN=MIDDLE  ALT=\"y%5E5+=+%282%5Ec%29%285%5Ed%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where c and d are multiples of 5<br><BR>\n" );
document.write( "Then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%282%5Ea%29%285%5Eb%29+=+5%282%5Ec%29%285%5Ed%29 ALIGN=MIDDLE  ALT=\"2%2A%282%5Ea%29%285%5Eb%29+=+5%282%5Ec%29%285%5Ed%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E%28a%2B1%29%29%285%5Eb%29+=+%285%5E%28d%2B1%29%29%282%5Ec%29 ALIGN=MIDDLE  ALT=\"%282%5E%28a%2B1%29%29%285%5Eb%29+=+%285%5E%28d%2B1%29%29%282%5Ec%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "From that we know<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+a%2B1 ALIGN=MIDDLE  ALT=\"c+=+a%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+d%2B1 ALIGN=MIDDLE  ALT=\"b+=+d%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For the smallest positive integer for which all those are true, we need to have c = d = 5; that makes a=4 and b=6.<br><BR>\n" );
document.write( "And we finally have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+%282%5E4%29%285%5E6%29 ALIGN=MIDDLE  ALT=\"x%5E2+=+%282%5E4%29%285%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%5E2+=+%282%5E5%29%285%5E6%29 ALIGN=MIDDLE  ALT=\"2x%5E2+=+%282%5E5%29%285%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E5+=+%282%5E5%29%285%5E5%29 ALIGN=MIDDLE  ALT=\"y%5E5+=+%282%5E5%29%285%5E5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  -->  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5y%5E2+=+%282%5E5%29%285%5E6%29 ALIGN=MIDDLE  ALT=\"5y%5E2+=+%282%5E5%29%285%5E6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the smallest positive integer that is both 2 times the square of one positive integer and 5 times the 5th power of another positive integer is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E5%29%285%5E6%29+=+500000 ALIGN=MIDDLE  ALT=\"%282%5E5%29%285%5E6%29+=+500000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500000+=+2%28250000%29+=+2%28500%5E2%29+=+2x%5E2 ALIGN=MIDDLE  ALT=\"500000+=+2%28250000%29+=+2%28500%5E2%29+=+2x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=500000+=+5%28100000%29+=+5%2810%5E5%29+=+5y%5E5 ALIGN=MIDDLE  ALT=\"500000+=+5%28100000%29+=+5%2810%5E5%29+=+5y%5E5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );