Seven consecutive integers are added.

The sum is $6$ times the greatest of the seven integers. what is the smallest integer

Find the smallest integer:

Consecutive integers are integers that follow each other in a fixed sequence.

Let's consider the $n$ be the smallest positive integer.

Then the seven consecutive integers are $n,n+1,n+2,n+3,n+4,n+5,n+6$.

Here $n+6$ is the greatest of the seven integers.

Find the smallest positive integer $n$.

The addition of seven consecutive integers is equal to the sum is $6$ times the greatest of the seven integers:

$\begin{array}{rcl}n+n+1+n+2+n+3+n+4+n+5+n+6& =& 6\left(n+6\right)\\ & \Rightarrow & 7n+21=6n+36\\ & \Rightarrow & 7n-6n=36-21\\ \therefore n& =& 15\end{array}$

Therefore, the smallest integer is$\mathbf{15}$.