Factorial Base 4

Number Theory Level 3

Let $N$ be the number of digits of $\large {\dfrac{(1000_4)!}{(332_4)!}}$ in base 4. Find $N$ .

Notations :

$!$ represents the factorial notation.
$a_b$ means $a$ base $b$ . The numbers in the problem above are in base 4.

This problem is from the Who Wants to Be A Mathematician competition.

The answer is 6.

2 solutions

Alex G
Apr 11, 2016

Relevant wiki: Number Base - Problem Solving

$\large {(1000_4)! = 1000_4 * 333_4 * (332_4)!}$

$\large {\frac{1000_4 * 333_4 * (332_4)!}{(332_4)!} = 1000_4 * 333_4 = 333000_4}$

$\large {\boxed {6}}$

Leonardo Vannini
May 12, 2016

$\frac { { (1000 }_{ 4 })! }{ { (332 }_{ 4 })! } =\frac { { (64 }_{ 10 })! }{ { (62 }_{ 10 })! } ={ (64\cdot 63) }_{ 10 }<{ (64\cdot 64) }_{ 10 }={ 4 }_{ 10 }^{ 6 }={ 1000000 }_{ 4 }$ hence $\frac { { (1000 }_{ 4 })! }{ { (332 }_{ 4 })! } <{ 1000000 }_{ 4 }$ so N=6

Factorial Base 4

This problem is from the Who Wants to Be A Mathematician competition.

The answer is 6.

2 solutions

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