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Powered Pair (Posted on 2013-04-12) Difficulty: 2 of 5
Find all possible pairs (a,b) of positive integers such that aa has b digits and bb has a digits, where each of aa and bb is expressed in base ten.

No Solution Yet Submitted by K Sengupta    
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Solution Integers only, please (spoiler) | Comment 1 of 2
Without loss of generality, assume a <= b
Then b = length(aa) <= length(b) = a
So b <= a
But a <= b, so b = a

We therefore need a^a having a digits

Well, the number of digits in a^a = 1 + [a*log(a)], (where [] is the floor function], so we need a-1<= a*log(a) < a

In other words, (a-1)/a <= log(a) < 1  (so a must be less than 10).  A quick check in excel finds that only 1, 8 and 9 work.

Final answer: (1,1), (8,8), (9,9)

Edited on April 12, 2013, 7:13 pm
  Posted by Steve Herman on 2013-04-12 14:34:20

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