document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023732>Question 1023732</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How to factorize X^4 + 1/X^4 – 3 by completing the square. I am getting (X^2+ 1/X^2)^2 - 5. However book's answer is (x^2+1/x^2)^2 +1)(x^2+1/x^2)^2 -1)\r<BR>\n" );
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document.write( "Thank you\r<BR>\n" );
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document.write( "waqar </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #639213 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=639213\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties: <BR>\n" );
document.write( "(a) are divisible by 5 or by 7 (inclusive or). <BR>\n" );
document.write( "(b) are divisible by 5. <BR>\n" );
document.write( "(c) are divisible by 7. <BR>\n" );
document.write( "(d) are not divisible by either 5 or 7. <BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "We will need the inclusive/exclusive principles.<BR>\n" );
document.write( "For example, we look for numbers divisible by 2 or 3 between 1 to 20.<BR>\n" );
document.write( "Using integer arithmetic, i.e. discard fractions from quotients, we know that there are 20/3=6 divisible by 3, and 20/2=10 divisible by 2.<BR>\n" );
document.write( "The quantity o f numbers divisible by 2 OR by 3 is NOT 6+10=16!<BR>\n" );
document.write( "Why?  It's because 6 is divisible by both two and three, and its multiples have been counted twice.  So we must subtract 20/6=3 from 16 to get the right answer, namely 13, or 6+10-3=13, using the inclusion/exclusion principle.<BR>\n" );
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document.write( "Now for the pool of numbers between 1000 and 9999, we calculate the following:<BR>\n" );
document.write( "(a) are divisible by 5 or by 7 (inclusive or). <BR>\n" );
document.write( "Divisible by 5: 9999/5-1000/5=1999-200=1799<BR>\n" );
document.write( "Divisible by 7: 9999/7-1000/7=1428-142=1286<BR>\n" );
document.write( "Divisible by 35: 9999/35-1000/35=285-28=257<BR>\n" );
document.write( "So <BR>\n" );
document.write( "Divisible by 5 OR 7 = 1799+1286-257=2828<BR>\n" );
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document.write( "(b) are divisible by 5. <BR>\n" );
document.write( "See part (a) \r<BR>\n" );
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document.write( "(c) are divisible by 7. <BR>\n" );
document.write( "See part (a)<BR>\n" );
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document.write( "(d) are not divisible by either 5 or 7. <BR>\n" );
document.write( "The pool of numbers is from 1000 to 9999, namely 9000 numbers, out of which 2828 are divisible by either 5, or 7 or both.  The difference of 9000 and 2828 would therefore be those that are divisible neither by 5 nor 7.<BR>\n" );
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