Let the 5 integers be a,b,c,d and e (arranged in ascending order).
Since we know that the median is five, we can conclude that the value of c=5.
It's given that the mode is 1. 1<5, and has to be repeated more than once. Therefore, a and b = 1.
Now, the mean is 4. So, (1+1+5+d+e)=4
d+e=13 (d and e have to be greater than 5)
Which makes d=6 and e=7
In conclusion, the 5 integers are 1,1,5,6,7