Question

If $n$is a positive integer,prove that there exist $n$consecutive composite integers.

Short answer

The given statement is proved.

Step-by-step solution

Define  n

Consider the $n$are consecutive composite integer.

Now assume that,

$\left(n+1\right)!+2,\left(n+1\right)!+3,........\left(n+1\right)!+\left(n+1\right)$

As $k$exists such that $2\le k\le n+1$ that divides $\left(n+1\right)!+k$ , above are the all-composite integers.

Each term in the sequence of consecutive integers is divided by some such$k$ .

Result

$\left(n+1\right)!+2,\left(n+1\right)!+3,........\left(n+1\right)!+\left(n+1\right)$

As$k$ exists such that $2\le k\le n+1$ that divides$\left(n+1\right)!+k$ , $n$ is composite integers. Hence it is proved.