3. Let $n$ be an integer greater than 1 . If $n$ is greater than 1200 times any one of its prime factors, find the smallest possible value of $n$.
(1 mark)
設 $n$ 為大於 1 的整數。若 $n$ 較它的任何一個質因數的 1200 倍為大, 求 $n$ 的最小可能值。
3. 3888
3. We consider the following cases.
- If $n$ is of the form $2^{a}$, then we have $n>2 \times 1200=2400$, and the smallest $n$ of this form is $2^{12}=4096$.
- If $n$ is of the form $3^{b}$, then we have $n>3 \times 1200=3600$, and the smallest $n$ of this form is $3^{8}=6561$.
- If $n$ is of the form $2^{a} 3^{b}$, then we have $n>3 \times 1200=3600$, and after some trial (for example, by first fixing $a$ ) we find that the smallest $n$ of this form is $2^{4} \times 3^{5}=3888$.
- If $n$ contains a prime factor greater than or equal to 5 , then we have $n>5 \times 1200=6000$. Combining the above cases, the answer is 3888 .