4 consecutive numbers mixed questions

looking at 4 consecutive numbers in various ways

some tasks
mostly for seeking general results
(not necessarily intended for students in this form)

question (12) is always an integer
justifying this is a demanding task but does not need to involve factorising a quartic expression:

for question (13) you can think about any four consecutive numbers (maybe write down several sets):
one must be a multiple of 2 (or 8)
one must be a multiple of 3 (or 12)
and one must be a multiple of 4 (or 8)

this is not easily demonstrated (for me anyway) using algebra
each starting number of the sequence of four must be 4n or 4n + 1 or 4n + 2 or 4n + 3
and each n must be 3k or 3k + 1 or 3k + 2

then consider the four resulting expressions:
[note that in the case where the starting number is a multiple of 4, the product must divide by 72]