document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112732>Question 112732</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the two factors of 15 with a difference of 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #82184 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=82184\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What you are looking for are two numbers, p and q, such that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Aq=15 ALIGN=MIDDLE  ALT=\"p%2Aq=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p-q=2 ALIGN=MIDDLE  ALT=\"p-q=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Actually, even though the problem doesn't say so, you are looking for integer values for p and q. \r<BR>\n" );
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document.write( "So what are the factors of 15? There are only two sets of integers p and q that satisfy <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Aq=15 ALIGN=MIDDLE  ALT=\"p%2Aq=15\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and those two sets are 1, 15 and 3, 5. \r<BR>\n" );
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document.write( "How do we know? Start dividing the number you want to factor by the prime numbers starting at 2 until you go past the square root of the number you want to factor. In this case, the square root of 15 is less than 4, so we know that, at most, we will have to divide 15 by 2, 3, and 5. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F2 ALIGN=MIDDLE  ALT=\"15%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, Not an integer, so 2 is NOT a factor.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2F3=5 ALIGN=MIDDLE  ALT=\"15%2F3=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so 3 and 5 are prime factors of 15. \r<BR>\n" );
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document.write( "We can see what will happen when we divide by 5 so we can skip that step. And we know that any number has itself and 1 as factors, so that completes the proof that the factors are: \r<BR>\n" );
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document.write( "1 and 15, or<BR>\n" );
document.write( "3 and 5 \r<BR>\n" );
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document.write( "Of those two sets of factors, only one set satisfies the second condition, that the difference be 2, and that is 3, 5. \r<BR>\n" );
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document.write( " \r<BR>\n" );
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document.write( "Ok, your turn. Find the two factors of 21 with a difference of 4. \r<BR>\n" );
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document.write( "John <BR>\n" );
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