## Task A-3.2.

Determine the smallest natural number that can be expressed in the form $50 a^{4}$ and in the form $3 b^{3}$ for some natural numbers $a$ and $b$.
## First Solution.

Let $m$ be the smallest natural number for which there exist $a, b \in \mathbb{N}$ such that $m$ is equal to

$$
50 a^{4}=3 b^{3}
$$

In the above equation, we see that the left side of the equation must be divisible by 3 because the right side is, which is only possible if the number $a^{4}$ is divisible by 3. Since 3 is a prime number, this is only possible if $a$ is divisible by 3.

The right side of the equation must be divisible by 2 and 5, which is similarly only possible if $b$ is divisible by 2 and 5.

Therefore, there exist natural numbers $a_{1}$ and $b_{1}$ such that $a=3 a_{1}$ and $b=10 b_{1}$.

1 point

Notice that $a$ and $b$ are the smallest possible when $a_{1}$ and $b_{1}$ are the smallest possible. Substituting into the initial equation, we get

$$
3^{3} a_{1}^{4}=2^{2} \cdot 5 b_{1}^{3}
$$

Similarly, we conclude that $a_{1}$ must be divisible by 2 and 5, and $b_{1}$ must be divisible by 3. Therefore, $a_{1}=10 a_{2}$ and $b_{1}=3 b_{2}$, for some $a_{2}, b_{2} \in \mathbb{N}$.

Again, we seek the smallest possible $a_{2}$ and $b_{2}$. The equation now becomes

$$
2^{2} \cdot 5^{3} a_{2}^{4}=b_{2}^{3}
$$

We conclude that $b_{2}=10 \cdot b_{3}$, for some $b_{3} \in \mathbb{N}$.

From the following equation $a_{2}^{4}=2 b_{2}^{3}$, we conclude that $a_{2}$ is of the form $2 a_{3}\left(a_{3} \in \mathbb{N}\right)$, and from the new equation $2^{3} a_{3}^{4}=b_{3}^{3}$, we conclude that $b_{3}$ is of the form $2 b_{4}\left(b_{4} \in \mathbb{N}\right)$. Finally, we get the equation $a_{3}^{4}=b_{4}^{3}$, whose smallest solution in natural numbers is $a_{3}=b_{4}=1$.

Therefore, the smallest natural number $m$ is

$$
m=50 a^{4}=50 \cdot\left(2^{2} \cdot 3 \cdot 5\right)^{4}=2^{9} \cdot 3^{4} \cdot 5^{6}=648000000
$$