document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=12200>Question 12200</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you please help to answer the following question:<BR>\n" );
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document.write( "   When a positive integer N is written in base 9, it is a two-digit number. When 6N is written in base 7, it is a three-digit number obtained from the two-digit number by writting digit 4 to its right. Find the decimal representationsof all such numbers N.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #6200 by </B><A HREF=/tutors/aboutme.mpl?userid=khwang><B>khwang(438)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=khwang><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=6200\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">     When a positive integer N is written in base 9, it is a two-digit number. When 6N is written in base 7, it is a three-digit number obtained from the two-digit number by writting digit 4 to its right. Find the decimal representationsof all such numbers N. [/quote]\r<BR>\n" );
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document.write( " Sol: By the given condtions, we have<BR>\n" );
document.write( " There are integers 1 <= a < 7, 0 <= b < 7 such that<BR>\n" );
document.write( " N = 9 a + b ... 1) <BR>\n" );
document.write( "6N = 7^2 a + 7 b + 4 ...(2)\r<BR>\n" );
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document.write( "By (1)*6: we have 6 N = 54 a + 6 b = 7^2 a + 5 a + 6 b = 7^2 a  + 7 b + 4.<BR>\n" );
document.write( " We have 5 a = b + 4 ...(3) \r<BR>\n" );
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document.write( "Note, 1 <= a <= 6, 0 <= b <= 6 and 5 a = b + 4.\r<BR>\n" );
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document.write( " Value of a starting from 1 ,consider the table below:\r<BR>\n" );
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document.write( " a    b(=5a-4)   N = 9 a + b   6N   (6N)base 7<BR>\n" );
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document.write( " 1	1        10          60     114 (OK)<BR>\n" );
document.write( " 2	6        24         144     264 (OK)<BR>\n" );
document.write( " 3    15-4 >6 (invalid)<BR>\n" );
document.write( " Similarly ,we see that b > 6 whenever 6>= a > 3,so no valid b if a > 3.\r<BR>\n" );
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document.write( " Thus, we obtain the two possible solutions 10 or 24.\r<BR>\n" );
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document.write( " [Double check : N = 10 [base 9] = 11, 6N = 60 , since 60 = 7^2 + 7 + 4,<BR>\n" );
document.write( " so 60[base 7] = 114. <BR>\n" );
document.write( " N = 24 [base 9] = 26, 6N = 144 , since 144 = 2*7^2 + 6*7 + 4,<BR>\n" );
document.write( " so 144[base 7] = 264. \r<BR>\n" );
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document.write( " This question may be not quite easy for you. Try to read carefully<BR>\n" );
document.write( " about every step.\r<BR>\n" );
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document.write( " Kenny<BR>\n" );
document.write( " </SPAN>\n" );
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