document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252284>Question 252284</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Each letter in the problem to the right represents a different digit<BR>\n" );
document.write( "All digits in NINE are odd. What is the smallest possible value of NINE?<BR>\n" );
document.write( "   FOUR<BR>\n" );
document.write( "  +FIVE<BR>\n" );
document.write( "_________<BR>\n" );
document.write( "   NINE\r<BR>\n" );
document.write( "\n" );
document.write( "a. 3135   b. 3537   c. 5153   d. 5351   e. 7173 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184300 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20060)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184300\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\"><font size = 4 color = \"indigo\"><b>\r\n" );
document.write( "Let a, b, c be the numbers \"to carry\", which\r\n" );
document.write( "can only be 0 or 1.\r\n" );
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document.write( "  a  b  c   \r\n" );
document.write( "  F  O  U  R\r\n" );
document.write( "+ F  I  V  E\r\n" );
document.write( "------------\r\n" );
document.write( "  N  I  N  E\r\n" );
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document.write( "in the leftmost column a=1\r\n" );
document.write( "because if a=0, then N=2F, \r\n" );
document.write( "making N even, but N is odd.\r\n" );
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document.write( "in the 2nd column from the left, \r\n" );
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document.write( "b + O + I = 10 + I\r\n" );
document.write( "b + O = 10\r\n" );
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document.write( "b can only be 1, for if b were 0,\r\n" );
document.write( "O would be 10, not a digit.\r\n" );
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document.write( "Thus O = 10-1 = 9\r\n" );
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document.write( "Now we have:\r\n" );
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document.write( "  1  1  c   \r\n" );
document.write( "  F  9  U  R\r\n" );
document.write( "+ F  I  V  E\r\n" );
document.write( "------------\r\n" );
document.write( "  N  I  N  E\r\n" );
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document.write( "From the rightmost column,\r\n" );
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document.write( "R + E = E + 10c\r\n" );
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document.write( "Subtracting E from both sides,\r\n" );
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document.write( "R = 10c\r\n" );
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document.write( "c can't be 1, for R would be 10,\r\n" );
document.write( "not a digit, so c=0.  Therefore\r\n" );
document.write( "R=0, and we have:\r\n" );
document.write( "\r\n" );
document.write( "  1  1  0\r\n" );
document.write( "  F  9  U  0\r\n" );
document.write( "+ F  I  V  E\r\n" );
document.write( "------------\r\n" );
document.write( "  N  I  N  E\r\n" );
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document.write( "E must be odd and less than 5, so as to \r\n" );
document.write( "carry only 0.  So we have all the \"carry\"'s,\r\n" );
document.write( "and E is either 1 or 3, so we have\r\n" );
document.write( "\r\n" );
document.write( "  1  1  0            1  1  0\r\n" );
document.write( "  F  9  U  0         F  9  U  0\r\n" );
document.write( "+ F  I  V  1   or  + F  I  V  3        \r\n" );
document.write( "------------       ------------ \r\n" );
document.write( "  N  I  N  1         N  I  N  3 \r\n" );
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document.write( "The largest odd value N could be is 7, \r\n" );
document.write( "since 9 has been used.  But if N were 7,\r\n" );
document.write( "then U+V would have to be 17, requiring \r\n" );
document.write( "U and V to be 9 and 8, but 9 has been \r\n" );
document.write( "used.  Therefore N is either 5 or 3.  \r\n" );
document.write( "But N can't be 3 because in either case\r\n" );
document.write( "above, that would make F be 1, and 1 has\r\n" );
document.write( "already been used.  So N can only be 5.\r\n" );
document.write( "\r\n" );
document.write( "  1  1  0            1  1  0\r\n" );
document.write( "  F  9  U  0         F  9  U  0\r\n" );
document.write( "+ F  I  V  1   or  + F  I  V  3        \r\n" );
document.write( "------------       ------------ \r\n" );
document.write( "  5  I  5  1         5  I  5  3 \r\n" );
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document.write( "and in both cases, F can only be 2, so\r\n" );
document.write( "now we have\r\n" );
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document.write( "  1  1  0            1  1  0\r\n" );
document.write( "  2  9  U  0         2  9  U  0\r\n" );
document.write( "+ 2  I  V  1   or  + 2  I  V  3        \r\n" );
document.write( "------------       ------------ \r\n" );
document.write( "  5  I  5  1         5  I  5  3 \r\n" );
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document.write( "Now the only possibilities for U and V are \r\n" );
document.write( "8 and 7, but we can't tell which is which.\r\n" );
document.write( "They could go either way, so we have four \r\n" );
document.write( "possibilities:\r\n" );
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document.write( "  1  1  0            1  1  0           1  1  0           1  1  0\r\n" );
document.write( "  2  9  7  0         2  9  8  0        2  9  7  0        2  9  8  0\r\n" );
document.write( "+ 2  I  8  1   or  + 2  I  7  1  or  + 2  I  8  3  or  + 2  I  7  3       \r\n" );
document.write( "------------       ------------      ------------      ------------\r\n" );
document.write( "  5  I  5  1         5  I  5  1        5  I  5  3        5  I  5  3    \r\n" );
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document.write( "All that's left is I, so we can easily fill that in in\r\n" );
document.write( "each case to make the addition correct, since all that's\r\n" );
document.write( "left for I in the the first two cases is 3 and 1 for \r\n" );
document.write( "the last two:\r\n" );
document.write( "\r\n" );
document.write( "  1  1  0            1  1  0           1  1  0           1  1  0\r\n" );
document.write( "  2  9  7  0         2  9  8  0        2  9  7  0        2  9  8  0\r\n" );
document.write( "+ 2  3  8  1   or  + 2  3  7  1  or  + 2  1  8  3  or  + 2  1  7  3       \r\n" );
document.write( "------------       ------------      ------------      ------------\r\n" );
document.write( "  5  3  5  1         5  3  5  1        5  1  5  3        5  1  5  3\r\n" );
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document.write( "So those are the only four possibilities, and the smallest\r\n" );
document.write( "value for NINE is 5153 in the last two cases.  So the correct \r\n" );
document.write( "choice is c. 5153.\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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