1. The members $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of a geometric sequence are natural numbers less than 2008. The number $a_{2}$ is divisible by 5, $a_{3}$ is divisible by 4, $a_{4}$ is divisible by 3, and the number $a_{1}$ is not divisible by 6. No prime number divides all 5 members of the sequence. Calculate the members of this sequence.
IV/1. Let $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ be terms of a geometric sequence, which can be written in the form $a_{i}=a_{1} \cdot q^{i-1}$ for $i=2,3,4,5$, where $q$ is some real number. However, $q=\frac{a_{2}}{a_{1}}$ is the quotient of two natural numbers, hence it is a rational number. Let's write $q=\frac{m}{n}$ as a reduced fraction. Then the terms of the sequence are

$$
a_{1}, \frac{a_{1} \cdot m}{n}, \frac{a_{1} \cdot m^{2}}{n^{2}}, \frac{a_{1} \cdot m^{3}}{n^{3}}, \frac{a_{1} \cdot m^{4}}{n^{4}}
$$

Since all terms are natural numbers and $m$ and $n$ are coprime, $n^{4}$ must be a divisor of $a_{1}$. Therefore, we can write $a_{1}=d n^{4}$, where $d$ is some natural number. Thus, the terms of the sequence are

$$
d n^{4}, d m n^{3}, d m^{2} n^{2}, d m^{3} n, d m^{4}
$$

Since there is no prime number that divides all terms of the sequence, it follows that $d=1$, and the sequence is of the form $n^{4}, m n^{3}, m^{2} n^{2}, m^{3} n, m^{4}$. We also know that all terms are less than 2008, so $m^{4}<2008$ and $n^{4}<2008$. From this, it follows that $m \leq 6$ and $n \leq 6$. However, the term $a_{1}=n^{4}$ is not divisible by 6, so $n \leq 5$. We also know that $a_{2}=n^{3} m$ is divisible by 5, $a_{3}=n^{2} m^{2}$ is divisible by 4, and $a_{5}=m^{3} n$ is divisible by 3, so the product $m n$ is divisible by 2, 3, and 5, hence by 30. At the same time, $m n \leq 6 \cdot 5=30$, so the product is exactly 30, from which it follows that $m=6$ and $n=5$. The terms of the sequence are thus $5^{4}=625$, $5^{3} \cdot 6=750$, $5^{2} \cdot 6^{2}=900$, $5 \cdot 6^{3}=1080$, and $6^{4}=1296$.
Conclusion that $q \in \mathbb{Q}$ ..... 2 points
Observation that $n^{4} \mid a_{1}$ ..... 1 point
Observation that $d=1$ ..... 1 point
Limitation $m, n \leq 6$ ..... 1 point
Conclusion $m=6, n=5$ and writing the sequence ..... 2 points