Alice the number theorist knows the rule for testing if a number n is divisible by 3:
n is divisible by 3 if and only if the sum of the digits of n is divisible by 3.
When Alice visits Mars, she finds that the Martians have six hands, and six fingers on each
hand, so that they count in base 36. In base 36, Alice’s divisibility test doesn’t work for testing
divisibility by d = 3. But it does work for one of the d listed below. Which one? Pick.
A) 4
B) 7
C) 10
D) 11
E) 15
First of all, why does the rule work in base 10 for 3? It works because 10-1 = 3*3,   100 -1 = 3 *33, 1000 - 1 = 3 * 333, etc. As a result, the expressions   a0 + a1 *10 + a2 /102 + a3 /103 + ... + ak *10k   and   a0 + a1 + a2 + a3 + ...+ ak   differ by a multiple of 3.   The key point is that 3 |(10 - 1). It is quite easy to see that this is a necessary condition. We   can see that it is sufficient because 100-1 = (10-1) *(10+1), 1000-1 = (10*1)(100+   10 + 1), etc. So, in base 36, we must have d | (36 - 1). Therefore, d must be a divisor of 35,   and the only such divisor listed is 7.   correct answer 7
<result>B</result>
Alice the number theorist knows the rule for testing if a number n is divisible by 3:
n is divisible by 3 if and only if the sum of the digits of n is divisible by 3.
When Alice visits Mars, she finds that the Martians have six hands, and six fingers on each
hand, so that they count in base 36. In base 36, Alice’s divisibility test doesn’t work for testing
divisibility by d = 3. But it does work for one of the d listed below. Which one?