I presume the 2nd one refers to 'consecutive integer'.
Let us set our numbers as x (divisible by 3), x+1 and x+2. For example, 3, 4 and 5.
x+1 + x+2 = 2x+3, divisible by 3. However, any sum involving the multiple of 3 cannot be divisible by 3, because you end up with either 2x+1 or 2x+2.
Re: the first one,
Basically it's the same answer as the 2nd one. Taking 5 consecutive numbers, one MUST be divisible by 5, and thus so must the product. It's fairly clear where this assertion comes from.