x^2 + dx + 72 = 0 has two distinct integer roots; how many values are possible for d?
Pick: A) 3 B) 6 C) 8 D) 12 E) 24
For a quadratic equation ax^2+dx+c=0, we know that -d/a is sum of roots and c/a is product of roots.   The quadratic equation here is x^2 + dx + 72 = 0, where product of roots is 72.   If we find all the factors of 72, we have the answer.   By prime factorization, we get 72= 2^3*3^2.   We know that total factors are  (3+1)*(2+1)= 
<gadget id="calculator"> (3+1)*(2+1)</gadget>
<output>12</output>
12 (Reason: with 2^n, we have n+1 possibilities. n^0 to n^n. so n+1)=12
<result>D</result>
x^2 + dx + 72 = 0 has two distinct integer roots; how many values are possible for d?