What Is the Missing Digit in $2^{29}$?

[anemone]

Here is this week's POTW:

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The number $2^{29}$ has exactly 9 distinct digits. Find the missing digit.

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[anemone]

Congratulations to the following members for their correct solution:):

1. kaliprasad
2. castor28

Solution from castor28:

Spoiler

Since $2^6 = 64\equiv1\pmod9$ and $29\equiv5\pmod6$, we have $2^{29}\equiv 2^5=32\equiv5\pmod9$.

We could also use the casting out nines technique: since $10\equiv1\pmod9$, this shows that the sum of the digits of the number is congruent to 5 modulo 9.

The sum of the numbers $0\ldots 9$ is $45$. Therefore, if $2^{29}$ consists of nine different digits, the missing digit must be $(45-5)\bmod9 = 4$.