document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006072>Question 1006072</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a, b, c are odd integers, then the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ax%5E2+%2B+bx+%2B+c+=+0+ ALIGN=MIDDLE  ALT=\"+ax%5E2+%2B+bx+%2B+c+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > has no integer solution.\r<BR>\n" );
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document.write( "Is it possible to prove this with an expression? <BR>\n" );
document.write( "E.g. product of 2 even number is even would be. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2n%2A2m=4nm+ ALIGN=MIDDLE  ALT=\"+2n%2A2m=4nm+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Is it possible to do it for the question above?\r<BR>\n" );
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document.write( "Im really sorry if its the wrong section </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622229 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=622229\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's assume that there does exist some integer solution x that makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+ax%5E2+%2B+bx+%2B+c+=+0+ ALIGN=MIDDLE  ALT=\"+ax%5E2+%2B+bx+%2B+c+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" > true. We'll show there's a contradiction.\r<BR>\n" );
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document.write( "Break it down into 2 cases:\r<BR>\n" );
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document.write( "Case 1: x is an even integer\r<BR>\n" );
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document.write( "If x is even, then surely x^2 is even too (see proof 1 at the bottom of the page)\r<BR>\n" );
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document.write( "a is odd<BR>\n" );
document.write( "x^2 is even<BR>\n" );
document.write( "a*x^2 will be even since odd*even = even (see proof 2)\r<BR>\n" );
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document.write( "b is odd<BR>\n" );
document.write( "x is even<BR>\n" );
document.write( "b*x is even (same reason as above)\r<BR>\n" );
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document.write( "a*x^2 is even<BR>\n" );
document.write( "b*x is even<BR>\n" );
document.write( "a*x^2 + b*x is even (even + even = even; see proof 3)\r<BR>\n" );
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document.write( "a*x^2 + b*x is even<BR>\n" );
document.write( "c is odd<BR>\n" );
document.write( "a*x^2 + b*x + c is odd (even + odd = odd; see proof 4)\r<BR>\n" );
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document.write( "But 0 is an even number. This is because 0 = 2*0. So IF x was an even integer, then a*x^2 + b*x + c is odd making a*x^2 + b*x + c = 0 impossible to achieve. The result of a*x^2 + b*x + c has to be even.\r<BR>\n" );
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document.write( "So we have a contradiction here: we got an odd result even though we want the result to be even. So x can't be an even integer.\r<BR>\n" );
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document.write( "Case 2: x is an odd integer\r<BR>\n" );
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document.write( "If x is odd, then x^2 is odd (see proof 5)\r<BR>\n" );
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document.write( "a is odd<BR>\n" );
document.write( "x^2 is odd<BR>\n" );
document.write( "a*x^2 is odd (odd*odd = odd; see proof 6)\r<BR>\n" );
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document.write( "b is odd<BR>\n" );
document.write( "x is odd<BR>\n" );
document.write( "b*x is odd (odd*odd = odd; see proof 6)\r<BR>\n" );
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document.write( "a*x^2 is odd<BR>\n" );
document.write( "b*x is odd<BR>\n" );
document.write( "a*x^2 + b*x is even (odd + odd = even; see proof 7)\r<BR>\n" );
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document.write( "a*x^2 + b*x is even<BR>\n" );
document.write( "c is odd<BR>\n" );
document.write( "a*x^2 + b*x + c is odd (even + odd = odd; proof 4)\r<BR>\n" );
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document.write( "So we run into the same issue as before. a*x^2 + b*x + c is odd making a*x^2 + b*x + c = 0 impossible to happen if x were an odd integer\r<BR>\n" );
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document.write( "We get another contradiction. So x can't be odd either.\r<BR>\n" );
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document.write( "With integers, there are only two cases: an integer is either an even number or an odd number. There are no other casees.\r<BR>\n" );
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document.write( "In either case, a*x^2 + b*x + c is going to be odd if a,b,c are odd integers and x is any integer. \r<BR>\n" );
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document.write( "This is where the contradiction occurs. If we assume there is an integer solution, then a*x^2 + b*x + c is odd when we want it to be even. \r<BR>\n" );
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document.write( "This proves that a*x^2 + b*x + c = 0 has no integer solutions if a,b,c are odd integers.\r<BR>\n" );
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document.write( "This ends the main proof. Below are the auxilary theorems used and their corresponding proofs.\r<BR>\n" );
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document.write( "Claims and proofs\r<BR>\n" );
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document.write( "Claim 1: if x is an even integer, then so is x^2\r<BR>\n" );
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document.write( "Proof 1: <BR>\n" );
document.write( "x is an even integer, so x = 2k for some integer k\r<BR>\n" );
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document.write( "x = 2k<BR>\n" );
document.write( "x^2 = (2k)^2<BR>\n" );
document.write( "x^2 = 4k^2<BR>\n" );
document.write( "x^2 = 2*2k^2<BR>\n" );
document.write( "x^2 = 2*m ... m is an integer, m = 2k^2<BR>\n" );
document.write( "that proves x^2 is even as well\r<BR>\n" );
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document.write( "---------------------------\r<BR>\n" );
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document.write( "Claim 2: odd*even = even \r<BR>\n" );
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document.write( "Proof 2:<BR>\n" );
document.write( "let x be an odd integer, y be an even integer<BR>\n" );
document.write( "x = 2k+1<BR>\n" );
document.write( "y = 2m<BR>\n" );
document.write( "k,m are integers\r<BR>\n" );
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document.write( "x*y = (2k+1)*2m<BR>\n" );
document.write( "x*y = 4km+2m<BR>\n" );
document.write( "x*y = 2*(2km+m)<BR>\n" );
document.write( "x*y = 2*(some integer)<BR>\n" );
document.write( "claim is confirmed\r<BR>\n" );
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document.write( "Note: when I say \"odd*even\" I mean \"odd integer times even integer\". It's my way of doing it shorthand.\r<BR>\n" );
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document.write( "---------------------------\r<BR>\n" );
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document.write( "Claim 3: even + even = even\r<BR>\n" );
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document.write( "Proof 3:\r<BR>\n" );
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document.write( "let x,y be two even integers<BR>\n" );
document.write( "x = 2k<BR>\n" );
document.write( "y = 2m<BR>\n" );
document.write( "k,m are integers\r<BR>\n" );
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document.write( "x+y = 2k+2m = 2*(k+m) = 2*(some integer)<BR>\n" );
document.write( "so x+y is even<BR>\n" );
document.write( "that proves the claim even + even = even\r<BR>\n" );
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document.write( "Claim 4: even + odd = odd\r<BR>\n" );
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document.write( "Proof 4: <BR>\n" );
document.write( "x is even, y is odd<BR>\n" );
document.write( "x = 2k, y = 2m+1<BR>\n" );
document.write( "k,m are integers\r<BR>\n" );
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document.write( "x+y = 2k+2m+1<BR>\n" );
document.write( "x+y = 2(k+m)+1<BR>\n" );
document.write( "x+y = some odd integer<BR>\n" );
document.write( "claim is confirmed\r<BR>\n" );
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document.write( "---------------------------\r<BR>\n" );
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document.write( "Claim 5: If x is odd, then x^2 is odd\r<BR>\n" );
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document.write( "Proof 5: \r<BR>\n" );
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document.write( "x is odd<BR>\n" );
document.write( "x = 2k+1, k is any integer<BR>\n" );
document.write( "x^2 = (2k+1)^2<BR>\n" );
document.write( "x^2 = 4k^2 + 4k + 1<BR>\n" );
document.write( "x^2 = 2*(2k^2 + 2k) + 1<BR>\n" );
document.write( "x^2 = 2*m + 1 .... m is an integer, m = 2k^2+2k<BR>\n" );
document.write( "so x^2 is odd<BR>\n" );
document.write( "claim is confirmed\r<BR>\n" );
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document.write( "---------------------------\r<BR>\n" );
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document.write( "Claim 6: odd*odd = odd\r<BR>\n" );
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document.write( "Proof 6: \r<BR>\n" );
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document.write( "Let x,y be two odd integers<BR>\n" );
document.write( "x = 2k+1, y = 2m+1 where k,m are integers\r<BR>\n" );
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document.write( "x*y = (2k+1)*(2m+1)<BR>\n" );
document.write( "x*y = 4km + 2k + 2m + 1<BR>\n" );
document.write( "x*y = 2*(2km + k + m) + 1<BR>\n" );
document.write( "x*y = 2*n + 1 ... where n is an integer<BR>\n" );
document.write( "this proves that x*y is odd\r<BR>\n" );
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document.write( "Claim 7: odd + odd = even\r<BR>\n" );
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document.write( "Proof 7: <BR>\n" );
document.write( "x,y are two odd integers<BR>\n" );
document.write( "x = 2k+1<BR>\n" );
document.write( "y = 2m+1<BR>\n" );
document.write( "k,m are integers\r<BR>\n" );
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document.write( "x+y = (2k+1)+(2m+1)<BR>\n" );
document.write( "x+y = (2k+2m)+(1+1)<BR>\n" );
document.write( "x+y = 2k+2m+2<BR>\n" );
document.write( "x+y = 2(k+m+1)<BR>\n" );
document.write( "x+y = 2*n ... n is an integer; n = k+m+1<BR>\n" );
document.write( "so x+y is even; the claim has been confirmed </SPAN>\n" );
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