×

## Number Bases

How would math be different if humans only had 6 fingers and 8 toes? How does a computer do math if it has no fingers to count on?

# Level 3

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 a 3-digit integer such that it is equal to $$100x+10y+z$$ for $$x,y,z$$ are non-negative integers and $$0 \le x,y,z \le 9$$.

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.

$$N$$ 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?

$1,3,4,9,10,12,13,\ldots$

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.

×