document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196802>Question 1196802</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If a number of two digits is divided by the sum of its digits, the quotient is 4 and the remainder is 6. If 9 is added to the number the sum has the same digits but inverted. What is the number?  </SPAN>\n" );
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document.write( "<font color=red>Answer: 34</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "a = tens digit<BR>\n" );
document.write( "b = units digit<BR>\n" );
document.write( "Each 'a' and b value is selected from the set {0,1,2,3,4,5,6,7,8,9} where repeats are possible.\r<BR>\n" );
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document.write( "The two digit number is of the form 10a+b<BR>\n" );
document.write( "For example, if a = 7 is the tens digit and b = 5 is the units digit, then 10a+b = 10*7+5 = 75\r<BR>\n" );
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document.write( "Divide this over the sum of the digits a+b and we get a quotient of 4 and remainder 6\r<BR>\n" );
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document.write( "(10a+b)/(a+b) = 4 remainder 6<BR>\n" );
document.write( "(10a+b)/(a+b) = 4 + 6/(a+b)<BR>\n" );
document.write( "10a+b = 4(a+b) + 6\r<BR>\n" );
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document.write( "Let's solve for b<BR>\n" );
document.write( "10a+b = 4(a+b) + 6<BR>\n" );
document.write( "10a+b = 4a+4b + 6<BR>\n" );
document.write( "b-4b = 4a+6-10a<BR>\n" );
document.write( "-3b = -6a+6<BR>\n" );
document.write( "b = (-6a+6)/(-3)<BR>\n" );
document.write( "b = 2a-2\r<BR>\n" );
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document.write( "If 9 is added to the original number 10a+b, then we're told the digits swap.\r<BR>\n" );
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document.write( "original + 9 = swapped digits<BR>\n" );
document.write( "(10a+b) + 9 = 10b+a\r<BR>\n" );
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document.write( "Now plug in b = 2a-2 and solve for 'a'<BR>\n" );
document.write( "(10a+b) + 9 = 10b+a<BR>\n" );
document.write( "(10a+2a-2) + 9 = 10(2a-2)+a<BR>\n" );
document.write( "12a+7 = 20a-20+a<BR>\n" );
document.write( "12a+7 = 21a-20<BR>\n" );
document.write( "12a-21a = -20-7<BR>\n" );
document.write( "-9a = -27<BR>\n" );
document.write( "a = -27/(-9)<BR>\n" );
document.write( "a = 3\r<BR>\n" );
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document.write( "Lastly, let's find b<BR>\n" );
document.write( "b = 2a-2<BR>\n" );
document.write( "b = 2*3-2<BR>\n" );
document.write( "b = 6-2<BR>\n" );
document.write( "b = 4\r<BR>\n" );
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document.write( "The original number is 10a+b = 10*3+4 = 34\r<BR>\n" );
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document.write( "Let's divide this over the sum of the digits a+b = 3+4 = 7<BR>\n" );
document.write( "34/7 = 4 remainder 6<BR>\n" );
document.write( "So far so good\r<BR>\n" );
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document.write( "Side note: you can think of it like you have 34 cookies and 7 friends to pass them out to. <BR>\n" );
document.write( "Each friend gets 4 cookies so 7*4 = 28 cookies are taken so far. <BR>\n" );
document.write( "That means there are 34-28 = 6 cookies leftover.\r<BR>\n" );
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document.write( "Now add 9 to the number <BR>\n" );
document.write( "34+9 = 43<BR>\n" );
document.write( "and the digit swap part is confirmed as well.\r<BR>\n" );
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document.write( "The answer is fully confirmed.<BR>\n" );
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