document.write( "Question 1133241: Alice replaces each of the 2008 numbers 6, 7, 8, …, 2012, 2013 with the sum of
\n" ); document.write( "its digits. Brian replaces each of Alice’s numbers with the sum of its digits, and
\n" ); document.write( "Colin replaces each of Brian’s numbers with the sum of its digits. What is the
\n" ); document.write( "number which Colin obtains most frequently?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #750502 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "A)  After Alice completed her operations, she obtained 2008 numbers A(n) ranged from 1 to 28 inclusively \r\n" );
document.write( "\r\n" );
document.write( "    (notice that 1 is the sum of digits of the number 10, while 28 is the maximum of the sums of the digits to each \r\n" );
document.write( "\r\n" );
document.write( "     of the numbers from 6 to 2013; namely, 28 is the sum of digits of the number 1999).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Again, when Alice completed her operations, she obtained 2008 numbers A(n) each of which is between 1 and 28 inclusively. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "B)  After Brian completed his operations, he obtained 2008 numbers B(n) ranged from 1 to 10 inclusively \r\n" );
document.write( "\r\n" );
document.write( "    (notice that 10 is the sum of digits of the number 28 from n.A) above).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Again, when Brian completed his operations, he obtained 2008 numbers B(n) each of which is between 1 and 10 inclusively. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "C)  After Colin completed his operations, he obtained 2008 numbers C(n) ranged from 1 to 9 inclusively \r\n" );
document.write( "\r\n" );
document.write( "    (notice that 9 is the sum of digits of the number 9).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Again, when Colin completed his operations, he obtained 2008 numbers C(n) each of which is between 1 and 9 inclusively. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "          It gives me an idea to consider the numbers at each step by the modulo of 9.\r\n" );
document.write( "\r\n" );
document.write( "          The second idea is that taking the sum of the digits of any integer number gives the number which has the same remainder \r\n" );
document.write( "                                                                                \r\n" );
document.write( "          when divided by 9  as the original number n has.  //  It is well known \"rule of divisibility by 9\" in the extended form.\r\n" );
document.write( "\r\n" );
document.write( "          Combination of these two ideas provides the solution as follows.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At step A), each number n    modulus 9 is equal to A(n) modulus 9:           n (mod 9) = A(n) (mod 9).\r\n" );
document.write( "\r\n" );
document.write( "At step B), each number A(n) modulus 9 is equal to B(A(n)) modulus 9:        n (mod 9) = A(n) (mod 9) = B(A(n)) (mod 9).\r\n" );
document.write( "\r\n" );
document.write( "At step C), each number B(A(n)) modulus 9 is equal to C(B(A(n))) modulus 9:  n (mod 9) = A(n) (mod 9) = B(A(n)) (mod 9) = C(B(A(n)) (mod 9).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus finally  n (mod 9) = C(n) (mod 9),\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "or, in other words, after step C), the obtained number C(n) has the same remainder when divided by 9 as the original number n has.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that the final sequence of numbers C(n) is sequential row of remainders n (mod 9), starting from 6 and ending by 2013 (mod 9) = 6:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus this row is  6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, . . . . , 1, 2, 3, 4, 5, 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So cycles of the length 9 from 6 to 5 are repeating cycles, and the last number \"6\" is the  first  AND THE LAST number  after the last loop.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The number which Colin obtains most frequently is \"6\".\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "            *************\r
\n" ); document.write( "\n" ); document.write( "              S O L V E D\r
\n" ); document.write( "\n" ); document.write( "            *************\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );