document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1023860>Question 1023860</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A 3 digit number is equal to 17 times the sum of its digits. If 198 is added to the number the digits are interchanged.The addition of 1st and 3rd digit is 1 less than middle digit. find the number </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "A 3 digit number is equal to 17 times the sum of its digits. If 198 is added to the number the digits are interchanged.The addition of 1st and 3rd digit is 1 less than middle digit. find the number<BR>\n" );
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document.write( "Solution:<BR>\n" );
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document.write( "There may be easier ways to solve this problem, but here's one solution.<BR>\n" );
document.write( "Clearly there are three variables (but I am not keen on solving a 3x3 system of linear equations), and we can reason as follows.<BR>\n" );
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document.write( "Let the number be \"xyz\", i.e. the numeric value is 100x+10y+z where x,y,z are the hundreds, tens and unit digits respectively.<BR>\n" );
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document.write( "The sum of first and last digits is 1 less than the middle digit means<BR>\n" );
document.write( "x+z=y+1...............(1)\r<BR>\n" );
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document.write( "If 198 is added to the number, the digits are \"interchanged\" or reversed, so<BR>\n" );
document.write( "100x+10y+z+198=100z+10y+x<BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "99(z-x)=198, or<BR>\n" );
document.write( "z-x=2..............(2)<BR>\n" );
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document.write( "Add (1) and (2) to get a simpler equation:<BR>\n" );
document.write( "2z=y+1................(3)<BR>\n" );
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document.write( "From (3) we can make a possible list of values relating y and z, for example, when y=1, 2z=1+1, so z=1, etc.<BR>\n" );
document.write( "(y,z)={(1,1),(3,2),(5,3),(7,4),(9,5)}<BR>\n" );
document.write( "noting the fact that y cannot take on even values or else x will not be an integer.<BR>\n" );
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document.write( "Now that we have limited our solutions to five, it's time to substitute values of y and z into equation (2) to find x.  For example, z=1 => x=-1 so reject negative values.  Again, z=2, x=0, that makes a two digit number, so reject again.  Try z=3, then x=1...<BR>\n" );
document.write( "(x,y,z)={(1,5,3),(2,7,4),(3,9,5)}<BR>\n" );
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document.write( "So the candidates are {153, 274, 395}<BR>\n" );
document.write( "From this list, the number(s) that is(are) divisible by 17 will fit the bill!<BR>\n" );
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