Number Theory

Number Bases

Number Bases: Level 3 Challenges


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

Details and Assumption

xyz \overline{xyz} represents reading the digits together, instead of multiplying them out. For example, xyz9=81x+9y+z \overline{xyz}_9 = 81 x + 9y + z and zyx6=36z+6y+x \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.

NN is an integer whose representation in base bb is 777. Find the smallest positive integer bb for which NN is the fourth power of an integer.

(23)!=4032010=10011101100000002.(2^3)! = 40320_{10} = 1001\:1101\:1000\:0000_2.

The factorial shown above has seven trailing zeros.

How many trailing zeroes does the number (216)!(2^{16})! have in binary notation?


The above sequence consists of the powers of 33 or the sum of distinct powers of 33, arranged in increasing order.

Find the 100th100^\text{th} term of the sequence.


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