2. Find the smallest positive integer $a$, such that there exists a positive odd integer $n$, satisfying $2001 \mid \left(55^{n}+a \times 32^{n}\right)$.
（14th Irish Mathematical Olympiad）
(Given $2001=87 \times 23$, we get $87 \mid\left(55^{n}+a \times\right.$ $\left.32^{n}\right)$, and $231\left(55^{n}+a \times 32^{n}\right)$. Then $0 \equiv 55^{n}+a \times 32^{n} \equiv$ $(-32)^{n}+a \times 32^{n}=32^{n}(a-1)(\bmod 87)$. Hence $a-1=$ $0(\bmod 87)$. Similarly, we get $a+1 \equiv 0(\bmod 23)$. Let $a=87 k+$ $1(k \in \mathbf{N})$, then 23 । $(87 k+2)$. Thus, 23 । $(-5 k+2)$, i.e., $k_{\min }=5$. Therefore, $a_{\min }=436$.)