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Reverse Role (Posted on 2014-10-15) Difficulty: 3 of 5
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?

No Solution Yet Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution More Bases (spoiler) | Comment 2 of 4 |
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

  Posted by Steve Herman on 2014-10-15 12:03:10
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