When you're working with a Min/Max problem - such as this one that asks you to minimize the least value in a set of numbers - you will want to organize your work by asking yourself:
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1) Do the values have to be integers? (Here you're told that they do)
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2) Are zero and/or negative numbers possible? (Here they are not ruled out, so they are indeed possible.) Note that in most word problems negative numbers won't fit the situation (you cannot have -5 children, flowers, or sales), but when a Min/Max problem lacks the "story problem" aspect you will want to make sure you consider negatives.
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3) Can the numbers repeat? (Here, again, you're not told that they cannot, so the values can repeat)
Then if you're asked to minimize a value, you will want to maximize the others, and if you're asked to maximize a value, you should try to minimize the others (all while adhering to the constraints you've identified above).
You want to minimize the least value, so you will want to maximize the highest value. You're told that that value must be 16, and since there is no prohibition on repeating numbers you can maximize the other two values (you're working with "least," "greatest," and "two others") at 16. That means that your values are 16, 16, 16, and x.
Since the four values average to 8, that means that their sum is 32. Therefore 16 + 16 + 16 + x = 32. This then means that x = -16, making answer choice B correct.