Solution - Difficult Puzzle of Sum and Product

Wait, read the puzzle once! 

Let's remind that the numbers are greater than or equal to 2; means those can't be either 0 or 1.

Now take a look at what Sam & Pete says - 

Sam: I don’t know what the numbers are Pete.

Pete: I knew you didn’t know the numbers… But neither do I.

Sam: In that case, I do know the numbers.

If Sam was told 4 then straightway he would have numbers 2,2 in mind as 3,1 combination is invalid.

If teacher had told Sam 5 as a sum then too Sam had correct pair of numbers 2,3 immediately as 4,1 or 5,0 are invalid combinations. 

So Sam must have at least number 6. Valid combinations for this sum are (2,4), (3,3).

If it was (3,3) then Pete would had 9 & he would have identified this combination correctly as (9,1) is not valid combination. Since he too didn't know exact numbers, it must be some different combination.

And if teacher had told Pete 8 then too he would have easily figured out correct combination of (2,4) as (8,1) is not valid.

So Pete can't have number 1,2,3,4,5,6,7, 8 or 9 or 11.

Now if he had 10 then only possible combination (2,5) and he would have that immediately. So he wouldn't have made the statement that he too didn't know numbers.

Let's assume that he had number 12 as product. Now in this case valid combinations are (2,6), (3,4).  The sums of these valid combinations are 8 & 7 respectively.

Now depending on what sum the Sam had; he can identify the correct pair of numbers easily.