several different positive integers are written on a blackboard. the product of the smallest two of them is 16. the product of the largest 2 of them is 225. what is the sum of the integers?

44

Step-by-step explanation:

The integers 2,8,9 and 25 fit to the terms of the problem.

So the smallest two are 2 and 8 =>2*8=16

and 9*25=225

Note that another pair of integers which product is 16 can be 1 and 16.

However it means that both numbers which product is 225 are bigger  than 16 - this is not possible so 16*16=256>225

So only numbers 2,8,9 and 25 fit to the terms of the problem.

Note that there is no any integer number between 8 and 9 . So there are four integers in total and these integers are 2,8,9 and 25.

The sum of these integers is 2+8+9+25=44