Help...

[mc3003]

Prove that 7 to the power of 35 has some digit that appears at least 4 times in its decimal representation.

I've got the answer of 378818692265664781682717625943, but I don't know how to prove it.

[mithunsam]

This problem has nothing to do with GMAT. But, still I would answer...

This can be solved in many ways. One of the easiest ways is to use logarithms.

First we need to find how many digits 7^35 has.
For that, we have to take log7^35 = 35log7 ~ 29.6 = 30 digits.

We have a maximum of 10 digits (0 to 9).

For minimum number of digits, one possibility is that each digit occur 3 times (3*10 = 30). However, since we are multiplying only by 7, last digit cannot be 0 (a 10 is required), 5(a five is required) etc. So, atleast one digit has to occur more than 3 times (that is 4 times).

[jnelson0612]

While mithunsam is an incredibly impressive mathematician, this question does indeed have nothing to do with the GMAT.