Number Theory

Number Bases

Number Bases: Level 3 Challenges


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.

\(N\) is an integer whose representation in base \(b\) is 777. Find the smallest positive integer \(b\) for which \(N\) is the fourth power of an integer.

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

The factorial shown above has seven trailing zeros.

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


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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