$30 \cdot 34$ Let $S$ be the set of the sums of the squares of three consecutive integers. Then
(A) No element of $S$ is divisible by 2.
(B) No element of $S$ is divisible by 3, but some elements are divisible by 11.
(C) No element of $S$ is divisible by 3 or 5.
(D) No element of $S$ is divisible by 3 or 7.
(E) None of the above.
(21st American High School Mathematics Examination, 1970)
[Solution] Let the three consecutive integers be $n-1, n, n+1$. Therefore, each element in $S$ can be expressed as
$$
(n-1)^{2}+n^{2}+(n+1)^{2}=3 n^{2}+2 \text {. }
$$

When $n$ is even, it is an even number, so (A) is not true.
Moreover, no element in $S$ is divisible by 3 (each element leaves a remainder of 2 when divided by 3).
If $n=5 m+1$, then $3 n^{2}+2=5\left(15 m^{2}+6 m+1\right)$ is divisible by 5, so
(C) is not true.
If $n=7 m+2$, then $3 n^{2}+2=7\left(21 m^{2}+12 m+2\right)$ is divisible by 7, so
(D) is not true.

Furthermore, if $n=11 m+5$, then $3 n^{2}+2=11\left(33 m^{2}+30 m+7\right)$ is divisible by 11. Therefore, (B) is true.
Hence, the answer is (B).