Question

If s and t are integers greater than 1 and each is a factor of the integer n, which of the following must be a factor of n st ? I. \small s^{t} II. \small \left ( st \right )^{2} III. \small s+tA) NoneB) 1 onlyC) 2 onlyD) 3 onlyE) 1 and 2

Answer 1

A) None

Explanation:

1)
`s^t` shoudnt neccesarily be a factor of nst, for example, if s = 3, t = 4, and n = 12, then both s and t are factors of n, but
`3^4 = 81` is not a factor of nst = 144.

2)
`(st)^2` shoudnt neccesarily be a factor of nst. Let s be 4, let t be 6, and let n be 12. Then n is a factor of both s and t, but
`(st)^2 = 24^2` is not a factor of nst = 12*24. In fact, it is a greater number.

3) Again, s+t isnt necessarily a factor of nst, let s be 2 and t be 3. Then both s and t are factor of n = 12. However 5 = s+t is not a factor of nst = 72.

So, neither of the three options is guaranteed to be a factor of nst. In fact, for s = 4, t = 6, and n = 12, none of the three options are valid.