Problem 636

Suppose the first integer in the list is $n$.

Then the remaining four integers are $n+3, n+6, n+9$, and $n+12$.

Since the fifth number is a multiple of the first, then $\frac{n+12}{n}=\frac{n}{n}+\frac{12}{n}=1+\frac{12}{n}$ is an integer. Since $1+\frac{12}{n}$ is an integer, then $\frac{12}{n}$ is an integer, or $n$ is a positive divisor of 12 .

The positive divisors of 12 are $1,2,3,4,6$, and 12 , so there are 6 possible values of $n$ and so 6 different lists.

(We can check that each of 6 values of $n$ produces a different list, each of which has the required property.)

ANSwER: (D)