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You can put this solution on YOUR website!
\n" ); document.write( "Tutor @ikleyn didn't like the way the problem was stated, so she stated it in a different way and solved it. But her restated problem has four solutions (0, 2, 6, and 18); the original problem only has three solutions, because b=0 is not a valid base.
\n" ); document.write( "So here is a solution of the problem as it was posted.
\n" ); document.write( "For base b, we want the number 5b+6 to be divisible by b+6; that is, we want
\n" ); document.write( "
\n" ); document.write( "where k is an integer.
\n" ); document.write( "It is easy (with experience!) to find the values of b that make k an integer in that equation.
\n" ); document.write( "The general technique is to perform the indicated division to get a result that is an integer plus a remainder:
\n" ); document.write( "
\n" ); document.write( "For this expression to be an integer, (b+6) has to be a divisor of 24. And since b in this problem is a number base, b has to be 2 or greater.
\n" ); document.write( "If b is 2 or greater and b+6 is a divisor of 24, then b+6 is at least 8; the divisors of 24 that are 8 or greater are 8, 12, and 24, giving us three solutions to the problem:
\n" ); document.write( "(1) b+6 = 8 --> b = 2 --> k = 5-(24/8) = 5-3 = 2
\n" ); document.write( "Check: (5b+6)/(b+6) = 16/8 = 2
\n" ); document.write( "(2) b+6 = 12 --> b = 6 --> k = 5-(24/12) = 5-2 = 3
\n" ); document.write( "Check: (5b+6)/(b+6) = 36/12 = 3
\n" ); document.write( "(3) (b+6) = 24 --> b = 18 --> k = 5-(24/24) = 5-1 = 4
\n" ); document.write( "Check: (5b+6)/(b+6) = 96/24 = 4
\n" ); document.write( "So....
\n" ); document.write( "ANSWER: There are three bases in which (b+6) divides into (5b+6) without any remainder: 2, 6, and 18 \n" ); document.write( "