Composite Dilemma

Number Theory Level pending

Which of the following values is a possible value of $n$ such that $n, n+1, n+2,\ldots ,n+200$ are all composite?

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

Taken from my Algebra Book (Gallian).

2!+200

2002

202

200!+2

2 solutions

Hana Wehbi
May 14, 2016

Consider $n=200!+2$ .

Then $2$ divides $n$ ;

$3$ divides ( $n+1$ )

$4$ divides ( $n+2$ )... so on

and $202$ divides ( $n+200$ ).

Thus, we notice they are all composite since they have factors other than $1$ and the number itself.

Akash Patalwanshi
May 14, 2016

It is not solution. I had just eliminated three wrong options! By using the fact that, primes numbers are 'not' composite. Because they have "not" at least one positive divisior other than $1$ and $itself$ .

One can $easily$ see that, there are $primes$ between $2002$ to $2202$ , $202$ to $402$ , $2! + 200$ to $2! + 400$ . Because these numbers are small. So the three options are eliminated.

Composite Dilemma

Taken from my Algebra Book (Gallian).

2 solutions

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