Number Theory
# Number Bases

There is a number \(a\), which can be written as \(\overline{xyz}\) in base 9 and \(\overline{zyx}\) in base 6, for some positive integer \(x,y,z\). Find \(x+y+z\).

**Details and Assumption**

\( \overline{xyz} \) represents reading the digits together, instead of multiplying them out. For example, \( \overline{xyz}_9 = 81 x + 9y + z \) and \( \overline{zyx}_6 = 36 z + 6y + x \).

Find the smallest natural number greater than 3 which has the unit digit of 3 when expressed in base-4, base-5, base-6, base-7, and base-8.

Express your answer in base 10.

\[(2^3)! = 40320_{10} = 1001\:1101\:1000\:0000_2.\]

The factorial shown above has seven trailing zeros.

How many trailing zeroes does the number \((2^{16})!\) have in binary notation?

\[1,3,4,9,10,12,13,\ldots\]

The above sequence consists of the powers of \(3\) or the sum of distinct powers of \(3\), arranged in increasing order.

Find the \(100^\text{th}\) term of the sequence.

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