document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041791>Question 1041791</A>: <I><SPAN STYLE=\"word-break: break-all;\"> To show that is an irrational number unless n is a perfect square, explain how the assumption that is rational leads to a contradiction of the fundamental theorem of arithmetic by the following steps:<BR>\n" );
document.write( "(A) Assume that n is not a perfect square, that is, does not belong to the sequence 1, 4, 9, 16, 25, . . . . Explain why some prime number p appears an odd number of times as a factor in the prime factorization of n.<BR>\n" );
document.write( "(B) Suppose that where a and b are positive integers, Explain why<BR>\n" );
document.write( "(C) Explain why the prime number p appears an even number of times (possibly 0 times) as a factor in the prime factorization of<BR>\n" );
document.write( "(D) Explain why the prime number p appears an odd number of times as a factor in the prime factorization of<BR>\n" );
document.write( "(E) Explain why parts (C) and (D) contradict the fundamental theorem of arithmetic. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #656768 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=656768\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Fundamental Theorem of Arithmetic:  Every integer greater than 1 is either prime itself or the product of primes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Beyond this, I can't help you much because you have left stuff out of the questions.  I suspect you cut and paste from an on-line course and some of the information was graphical that did not copy and render.  Next time READ what you have in the dialog box BEFORE you hit send.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );