Lets start with option A

let \(x\)be the first integer than 2nd integer is\(x+1\)
There average will be \(\frac{2x + 1}{2} = x + \frac{1}{2}\) which is not a integer

For option B
1st integer = \(x\)
2nd integer = \(x+1\)
3rd integer = \(x+2\)

Hence there average \(= \frac{3x +3}{3}\) or, \(\frac{3(x+1)}{3} = x + 1\)
since x is an integer \(x +1\) is also a integer
option B
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Since the average of 3 consecutive integers is equal to the median, and since the median would be the middle integer of 3 consecutive integers, we see that the average of 3 consecutive integers must be an integer.

Answer: B
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