Since the prizes have to be identical, all the prizes should have the same number of popcorn packages, pretzel bags and DVDs.
So we need to find out: (A) how many such prizes can we make? (B) how many items will be in each of the prizes?
(A)
In order to do that, we need to distribute these items across the prizes. To know across how many prizes can be made, we need to find the highest common factor (HCF) across the numbers 56, 40 and 16.
Note: The highest common factor would be the largest of all the factors that are common to 56, 40 and 16. A factor is a number that even divides another number.
To find out the HCF, we will do prime factorization for the three numbers (i.e. break the number as a multiple of its prime factors):
56 = 2 x 2 x 2 x 7
40 = 2 x 2 x 2 x 5
16 = 2 x 2 x 2 x 2
From the above, we can see that the common factors across 56, 40 and 16 are 2 x 2 x 2. Hence, the HCF for 56, 40 and 16 is 8. (Please see the attached image to see the paper calculations, and T - chart method for finding out HCF)
Now we know we can make 8 prizes.
(B)
Next, to know how many popcorn packages, pretzel bags and DVDs will be included in each prize, we will divide the number of items by the HCF.
Hence, in each of the prizes we will have:
56 ÷ 8 = 7 popcorn packages
40 ÷ 8 = 5 pretzel bags
16 ÷ 8 = 2 DVDs
Hence, there would be 8 prizes, each with 7 popcorn packages, 5 pretzel bags and 2 DVDs