LCM equals Factorial

Number Theory Level 1

Find the sum of all natural numbers $n$ such that

$\text{lcm}(1,2,3,\ldots, n) = n!$

Notations:

$\text{lcm}(\cdot)$ denotes the lowest common multiple function.
$!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .

The answer is 6.

2 solutions

Seth Christman
Dec 5, 2016

The only possible answers for LCM to be $n!$ is if each term is relatively prime to each other. Since $4$ and $2$ are not relatively prime, $n <4$ . Thus, $n=1, 2, or 3$ and $1+2+3=6$

n=1 should not be allowed as LCM function must have at least 2 arguments for it to make sense, so the answer should be 5

Anirudh Sreekumar - 4 years, 6 months ago

The LCM function requires a finite set of non-zero integers. It is rather useless to use LCM of a one element set, but it is within the functions restrictions.

Seth Christman - 4 years, 6 months ago

Magnas Bera
Jul 16, 2019

we know that gcd(a,b)×lcm(a,b)=ab Lcm(1,2,3,..n)=n! So gcd(1,2,3..n)=1 Which is possible when n=3,n=2,n=1 Therefore answer=1+2+3=6

LCM equals Factorial

The answer is 6.

2 solutions

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