(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.

As 0.01234567890123456789... must be a base-10 fraction by definition (decimal point), the Wacko Calculator would need, in a manner, convert the values X_N and Y_N into base-10 in order to display the base-10 fraction. But, as 0.01234567890123456789... = (13717421/1111111111) with 13717421 being a composite (3607 * 3803), how should converting 13717421 into a different base make it prime?

Or is the 'decimal' point a 'base-N' point with the value of 0.abc equal to a*(N^-1)+b*(N^-2)+c*(N^-3)?

Posted by Dej Mar on 2010-08-28 21:48:38