Factorials And Divisors

Number Theory Level 2

$1!$ has 1 positive divisor.
$2!$ has 2 positive divisors.
$3!$ has 4 positive divisors.
$4!$ has 8 positive divisors.
$5!$ has 16 positive divisors.
$6!$ has $\text{\_\_\_\_\_\_}$ positive divisors.

Notation: $!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

The answer is 30.

1 solution

Arjen Vreugdenhil
Oct 22, 2016

Relevant wiki: Number of Factors

From the wiki above,

If $N = p_1 ^{q_1} p_2^{q_2} \ldots p_n ^ {q_n}$ , then $N$ has $d(N) = (q_1 +1) (q_2 +1) \cdots (q_n + 1)$ divisors.

With $6! = 720 = 2^4\cdot3^2\cdot 5.$

From this prime factorization we see immediately that there are $(4+1)(2+1)(1+1) = \boxed{30}$ positive divisors.

Very nice fake pattern. I was slightly surprised that 4! worked out!

Calvin Lin Staff - 4 years, 7 months ago

Essentially it is because $3+1 = (1+1)(1+1)$ .

Arjen Vreugdenhil - 4 years, 7 months ago

@Calvin Lin ,

Sir this question is very easy. You can see that. Then also it is marked with level 4.

One have to only find number of divisors of 720 which is a one line answer.

I posted one IMO question which was also rated level 4. This means that there is no difference between that IMO problem and this problem.

I don't think there is a deep reason behind this rating.

So, Please see and do the necessary changes.

Priyanshu Mishra - 4 years, 7 months ago

This is a brand new problem and ratings take some time to stabilize. It does seem like it's stabilizing around mid-level 3.

We were aggressive in featuring this problem (leading to many more people trying it), because I wanted to highlight the misconception in pattern recognition.

Calvin Lin Staff - 4 years, 7 months ago

Factorials And Divisors

The answer is 30.

1 solution

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