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