(A) Jason likes primes and he was excited by
Wacko Calculator. He has a calculator that allows him to add, subtract, multiply and divide in positive integer bases up to 36. Jason chose two base N positive integers X and Y, with N being a positive integer between 10 and 36 inclusively, where X and Y are relatively prime having the proviso that sod(X) is prime. Thereafter, he divided X by Y to obtain: .01234567890123456789.....
Determine the values of N for which this is possible.
(B) Keeping all the other conditions in (A) unaltered but disregarding the proviso that sod(X) is a prime number – Jason noted that there is precisely one value of N between 10 and 36 inclusively such that X is a prime number.
What is the value of N and what are the corresponding values of X and Y?
Note: sod(x) denotes the sum of digits of x.
The solutions below assume that the
sod function is working in the base being used. That is, for example, 5G in base17 when passed to the
sod function will return 5 + 16 = 21.
Part A)
N = 11, X = 51874849 (27311384 base11), Y = 5187484920 (2222222222 base11), sod(X) = 29
N = 14, X = 342313201 (336699CD base14), Y = 57850930995 (2B2B2B2B2B base14), sod(X) = 61
N = 19, X = 420512089 (8HFE184 base19), Y = 136245916840 (808080808 base19), sod(X) = 67
N = 24, X = 23971032501 (55AAFFKL base24), Y = 12680676193075 (4J4J4J4J4J base24), sod(X) = 101
N = 26, X = 45173470609 (5G6116H7 base26), Y = 28233419130675 (5555555555 base26), sod(X) = 59
Part B)
For the second part, I found 2 values of N that resulted in X being prime. Now, even being represented in a different base, if a value base10 is prime, it'll be prime in any base. The two values I came up with are:
N = 14, X = 342313201 (336699CD base14), Y = 57850930995 (2B2B2B2B2B base14)
N = 23, X = 85591965317 (123456789 base23), Y = 41426511213648 (MMMMMMMMMM base23)
So unless I'm missing something, these appear to be the desired solutions.

Posted by Justin
on 20100831 17:02:17 