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well.... weird question (certainly not from a real GMAT prep), but it has a clear answer (A).
st1 is sufficient. if k is a multiple of 3 then t and 12 has 3 as a common factor:
k/3 is integer n. so m/4+k/3 = m/4+n = (3m+12n)/12. hence t=3m+12n.
t is a multiple of 3 and so is 12.
st2 insuff: for example m=3 k=1 (then t=13 and has only 1 as common factor with 12). m=3 k=3 has more factors. hence insufficient.
note that the questions doesn't state that t and 12 have indeed 12 as common factor. it just looks for common factors of t and 12 that are not 1 or 12. in fact, if 12 is a common factor of any two numbers, so are 2,3,4 and 6.