Half Cost function-Mean Squared error

Dear All
In the regression of " M" samples, in order to evaluate an error
( Y real - Y predict ) the Cost function formula is used.
In some cases the Half Cost function or half mean squared is used.
The" 1/2M " is multiplying instead of "1/M" to the sums of squared (Yr- Yp)
If it is possible guide me please , the difference between two formulas
(Cost function and its Half) and how we've got the 1/2 coefficient.

Many appreciation
Iraj