If x is an integer less than or equal to −2 and y is an integer with a

x and y are integer and x≤-2, while |y|≥5, implying either y≤-5 or y≥5.
We are to determine whether the following statements must be true of xy.
I. xy is an integer. This is always true for all values of x and y which are integers. An integer multiplied by another integer always results in another integer. Hence I is always true.

II. xy is negative. Not always true. We know x≤-2, which is always a negative number, but y can be either positive or negative. When y≥5 then xy is negative, however, when y≤-5, then xy is positive. Hence II. is not always true.

III. |xy|≥10. This always true. we know x≤-2 and y is either ≤-5 or ≥5. in either case, the minimum possible value of |xy|=10, hence |xy|≥10 is always true.

Statements I and III are always true, hence the answer must be C.