Step-by-step explanation:
The Answer for the above question is
 The four integers are 1,2,5,10. And Sum of the four integers is 18 .
<u>Method : </u>
Product of four different positive integers is 100 .
<u>First</u> of all lets break the number "100"
  ⇒  100 = 50*2
  ⇒ 100 = 25 * 2 * 2
   ⇒ 100 = 5 * 5 * 2 * 2
But here we get the same  integers 2 and 5 and we want different positive integers . So hereby merge 5 * 2  which is 10. 
Let's get to the possible combination -
   After merging 5*2 = 10  we get the answer as -
  ⇒ 100 = 1 * 2 * 5 * 10 
  Here, we have 1, 2, 5, 10. these are the positive integers whose product gives 100 . 
 Sum of these products is -
    ⇒ Sum = 1 + 2 + 5 + 10
   ⇒  Sum = 18 .
Sum of these four Integers is 18.