K is a multiple of 24 but NOT a multiple of 16 which of the following cannot be an integer?
A) K/8 B) K/9 C) K/32 D) K/36 E) K/81

so, they explain two ways to do this. the second way makes sense to me (list multiples of 24, cross multiples of 16 off the list, and then just plug numbers to see what sticks) but the first explanation does not make sense.

can someone explain to me how to do this problem using prime factorization? I get that the prime factor or 24 is 2^3, 3^1 and that 16 is 2^4 but I fail to see how I solve the problem given this information.

its easy just understand thisâ€¦
K is a multiple of 24 but NOT a multiple of 16
K = 24 * (something) but not, 16 * (something)
what does this tells you
K = 2 * 2 * 2 * 3 * (something) but not, 2 * 2 * 2 * 2 * (something)
now look at answers every option is dividing K, so when will K be an integer ?
answer:-
a) 8 is completely gone when K / 8, so it will be an integer, this is not the answer as asked for non-integer value
b) 9 could be completely gone assuming (something = 3), question is asking for 100% possibility that K will not be an integer, so K / 9 might also give an integer.
c)32 = 16 * 2 = 2 * 2 * 2 * 2 * 2 = means since K will not have 2^4, denominator will still be left, means will give a non integer value, hence the answer.