document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=244665>Question 244665</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please Help<BR>\n" );
document.write( "what is the prime factorization of 288?<BR>\n" );
document.write( "It also tells me to  enter the smaller factor first.\r<BR>\n" );
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document.write( "Here's what I got<BR>\n" );
document.write( "2^3 x 6^2? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #178939 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=178939\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Sorry, not even close.  \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2^3\ \times\ 6^2\">\r<BR>\n" );
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document.write( "<i>is</i> a factorization of 288, BUT...\r<BR>\n" );
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document.write( "6 <i>is not</i> prime, therefore any factorization that contains a 6 is NOT a <i>prime</i> factorization.\r<BR>\n" );
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document.write( "Here's the process:\r<BR>\n" );
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document.write( "Start with the given number\r<BR>\n" );
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document.write( "288.  This number is even so it is divisible by 2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 288\ \div\ 2\ = 144\">, one factor of 2.\r<BR>\n" );
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document.write( "144.  This number is even so it is divisible by 2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 144\ \div\ 2\ = 72\">, two factors of 2.\r<BR>\n" );
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document.write( "72.  This number is even so it is divisible by 2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 72\ \div\ 2\ = 36\">, three factors of 2.\r<BR>\n" );
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document.write( "36.  This number is even so it is divisible by 2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 36\ \div\ 2\ = 18\">, four factors of 2.\r<BR>\n" );
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document.write( "18.  This number is even so it is divisible by 2.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 18\ \div\ 2\ = 9\">, five factors of 2.\r<BR>\n" );
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document.write( "9.  This number is NOT even so it is NOT divisible by 2.  This number is divisible by 3.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 9\ \div\ 3\ = 3\">, one factor of 3.\r<BR>\n" );
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document.write( "3 is prime.  Two factors of 3.\r<BR>\n" );
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document.write( "All together, 5 factors of 2 and 2 factors of 3\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2^5\,\cdot\,3^2\">\r<BR>\n" );
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document.write( "is the unique <i><b>prime</b></i> factorization of 288.\r<BR>\n" );
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document.write( "The illustrated process works for finding the prime factorization of any integer.  Always start with 2.  Numbers are divisible by 2 if they are even.  Once you arrive at an odd result, move on to 3.  If the sum of the digits is divisible by 3, the number is divisible by 3.  Once you get a result that is not divisible by 3, go on to the next higher prime number, namely 5.  Numbers that end in either 5 or 0 are divisible by 5.  7 doesn't have a convenient rule -- trial and error works well though.  If the center digit of a three digit number is the sum of the first and last digits, the number is divisible by 11, but you need to do trial and error anyway.  Continue in this pattern taking the next higher prime number as your divisor until you either clearly achieve a prime result or the next trial divisor (the next higher prime) is greater than the square root of the dividend.\r<BR>\n" );
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document.write( "For example, the prime factorization of 947.\r<BR>\n" );
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document.write( "Square root is a little bigger than 30.\r<BR>\n" );
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document.write( "Not even, so no factors of 2.<BR>\n" );
document.write( "Sum of digits is 20, 20 not divisible by 3, so no factors of 3.<BR>\n" );
document.write( "Does not end in 0 or 5, so no factors of 5.<BR>\n" );
document.write( "Calculator has a decimal fraction result when divide by 7, so no factors of 7.<BR>\n" );
document.write( "Same thing when dividing by 11, 13, 17, 19, 23, 29.<BR>\n" );
document.write( "The next prime number is 31 which is greater than the square root of 947\r<BR>\n" );
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document.write( "Conclusion: 947 is prime.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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