Question 1125943
mean of 5 with 7 integers can have total of 35.  But, a unique mode of 1 means at least 2 of those integers have to be 1.  The median of 6 means that 6 is the fourth from the bottom, and there are 3 integers higher than that.

The bottom 3 integers are 1,1, and one other.For the largest possible value of any integer, the third integer below the median has to be smaller or 1.

That makes 3 integers of 1 each, and 1 of 6.  The total is 35, so two of the remaining three are 7 and 8,  because there is a unique mode of 1.  That makes the total 24 (7,8,6,1,1,1).  The largest possible value is 11.