Posted 26 September 2007 - 09:11 PM

two possible answers here.

we realize that the number has to be at least 5 digits long, so we need to look for a power that has five digits...

3^9 = 19683,

which is too small.

the next one in the neighborhood is 4^8, which is also too small, and without even bothing to do the next multiplication, one can see that 4^9 would have too many digits.

that brings us to...

5^7 = 78125,

which is a correct answer. But is it the only one? 6^6 is slightly too small (and besides, one can make the reasonable assumption that different letters mean unique numbers). 7^5 is too small while 7^6 is too big. 8^5 is too small, and the next higher power will have too many digits. That brings us to the last possibility, which is 9^5 = 59049, which is a possibility, but only if you allow different letters to mean the same digit. I am assuming that such is not allowed, which would negate this option, so that means that 5^7 is the correct answer.