document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=565182>Question 565182</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find all values that satisfy ALL the following conditions: <BR>\n" );
document.write( "1. Positive whole numbers less than 100 <BR>\n" );
document.write( "2. Four more than each number is a multiple of 6. <BR>\n" );
document.write( "3. The sum of the digits of each number is a multiple of 4. <BR>\n" );
document.write( "4. Two digit numbers where the ten’s digit is greater than the one’s digit<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #365752 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=365752\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Modular arithmetic solution:\r<BR>\n" );
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document.write( "From (2) we know that the number is even, and it is congruent to -4 mod 6 (which is 2 mod 6 or 2 mod 3). Therefore the sum of the digits is 2 mod 3.\r<BR>\n" );
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document.write( "From (3), the sum of the digits is a multiple of 4. Since 8 is the only (small) multiple of 4 congruent to 2 mod 3, the sum of the digits is 8.\r<BR>\n" );
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document.write( "From (4), the tens digit is greater than the ones digit. Out of the numbers 17, 26, ..., 62, 71, 80, only the numbers 53, 62, 71, 80 satisfy (4). However we already know the number is even, so 62 and 80 are the only possible numbers. </SPAN>\n" );
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