Let S[x] be the digital root function (also known as the repeated digital sum function), where one adds the digits of positive integer x, then adds the digits of the sum until obtaining a single-digit number. (For example, S[975] = 3 because 9 + 7 + 5 = 21 and 2 + 1 = 3).

Given S[a^a] = 2, what is the smallest positive integer that a can be such that a is a perfect power?


Note: a is a perfect power if there exist natural numbers m > 1, and k > 1 such that m^k = a.

See previous post for earlier steps.

It wasn't hard to work out the smallest perfect power that equals 5 mod 18.  No power of 7, 13 or 17 ever equals 5 mod 18. 

The powers of 5 (mod 18) are 5, 7, 17, 13, 11, 1 and repeating thereafter with period 6.

The powers of 11 (mod 18) are 11, 13, 17, 7, 5, 1  and repeating thereafter with period 6.

So, the only candidates for minimum a are 5^7 and 11^5.

5^7 = 78125
11^5 = 161051

So, the answer (I think) is 78125.

I'm not, however, going to check it.

Nice problem, Dej Mar

Posted by Steve Herman on 2008-07-01 20:03:23