Question 1134705
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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.<br>
So here is a solution of the problem as it was posted.<br>
For base b, we want the number 5b+6 to be divisible by b+6; that is, we want<br>
{{{(5b+6)/(b+6) = k}}}<br>
where k is an integer.<br>
It is easy (with experience!) to find the values of b that make k an integer in that equation.<br>
The general technique is to perform the indicated division to get a result that is an integer plus a remainder:<br>
{{{(5b+6)/(b+6) = (5b+30-24)/(b+6) = (5b+30)/(b+6)-24/(b+6) = 5-24/(b+6)}}}<br>
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.<br>
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:<br>
(1) b+6 = 8  -->  b = 2  -->  k = 5-(24/8) = 5-3 = 2
Check: (5b+6)/(b+6) = 16/8 = 2<br>
(2)  b+6 = 12  -->  b = 6  -->  k = 5-(24/12) = 5-2 = 3
Check: (5b+6)/(b+6) = 36/12 = 3<br>
(3) (b+6) = 24  -->  b = 18  -->  k = 5-(24/24) = 5-1 = 4
Check: (5b+6)/(b+6) = 96/24 = 4<br>
So....<br>
ANSWER: There are three bases in which (b+6) divides into (5b+6) without any remainder: 2, 6, and 18