(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?
: 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 base-17 when passed to the sod
function will return 5 + 16 = 21.
N = 11, X = 51874849 (27311384 base-11), Y = 5187484920 (2222222222 base-11), sod(X) = 29
N = 14, X = 342313201 (336699CD base-14), Y = 57850930995 (2B2B2B2B2B base-14), sod(X) = 61
N = 19, X = 420512089 (8HFE184 base-19), Y = 136245916840 (808080808 base-19), sod(X) = 67
N = 24, X = 23971032501 (55AAFFKL base-24), Y = 12680676193075 (4J4J4J4J4J base-24), sod(X) = 101
N = 26, X = 45173470609 (5G6116H7 base-26), Y = 28233419130675 (5555555555 base-26), sod(X) = 59
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 base-10 is prime, it'll be prime in any base. The two values I came up with are:
N = 14, X = 342313201 (336699CD base-14), Y = 57850930995 (2B2B2B2B2B base-14)
N = 23, X = 85591965317 (123456789 base-23), Y = 41426511213648 (MMMMMMMMMM base-23)
So unless I'm missing something, these appear to be the desired solutions.
Posted by Justin
on 2010-08-31 17:02:17