Divisors of factorials I

Number Theory Level 2

What is the largest positive integer $m$ for which $2^m$ is a divisor of $9!$ ?

Notation: $!$ denotes the factorial notation . For example: $8!=1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$

The answer is 7.

1 solution

Chew-Seong Cheong
May 23, 2019

In general, the largest power $m$ of a prime $p$ that divides $n!$ is given by:

$\begin{aligned} m & = \sum_{k=1}^\infty \left \lfloor \frac n{p^k} \right \rfloor & \small \color{#3D99F6} \text{For }n = 9 \text{ and }p=2 \\ & = \left \lfloor \frac 92 \right \rfloor + \left \lfloor \frac 94 \right \rfloor + \left \lfloor \frac 98 \right \rfloor \\ & = 4 + 2 + 1 \\ & = \boxed 7 \end{aligned}$

Notation: $\lfloor \cdot \rfloor$ denotes the floor function .

@Hana Wehbi , the note for $!$ has been the standard description used in Brilliant.org for factorial. This is used in many other problems. It leads members to refer to the wikis in Brilliant.org. Hana, you should also refer to problems posted by others.

Chew-Seong Cheong - 2 years ago

@Chew-Seong Cheong Sir, How do l enter links in latex? This is something l don’t know how to do. Thank you.

Hana Wehbi - 2 years ago

In the editor, the second button of the menu bar with a chain icon is for inserting links. Or you can key in link name . Note that there is no space between ] and (.

Chew-Seong Cheong - 2 years ago

@Chew-Seong Cheong – Thank you, that was helpful.

Hana Wehbi - 2 years ago

Divisors of factorials I

The answer is 7.

1 solution

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