A two-digit base 7 positive integer has its digits reversed when expressed in base N.
What are the possible values of N satisfying the given conditions?
Let the digits be a and b
Then 7a + b = Nb + a
So N = 6a/b + 1
Without loss of generality, only consider a,b having no common factors
Then b can = 1,2,3 or 6
if b = 1, then N = 6a + 1. N must be > b, so N can be 7,13,19,25,31,37
if b = 2, then N = 3a + 1. N must be > b, so N can be 4,7,10,13,16,19
if b = 3, then N = 2a + 1. N must be > b, so N can be 5,7,9,11,13
if b = 6, then N = a + 1. N must be > b, so N can be 7
Summarizing, N can be 4,5,7,9,10,11,13,16,19,25,31,37
For instance, N = 25 when b = 1 and a = 4
14 (base 25) = 41 (base 7) = 29