## Task A-1.6.

Determine the smallest natural number $n$ such that half of $n$ is the square of some natural number, a third of $n$ is the cube of some natural number, and a fifth of $n$ is the fifth power of some natural number.
## Solution.

Let $n$ be the desired number. Since $n$ is the smallest number with the described properties, and we know it is divisible by 2, 3, and 5, it must be of the form $n=2^{a} \cdot 3^{b} \cdot 5^{c}$, where $a, b, c$ are natural numbers.

Half of this number is $2^{a-1} \cdot 3^{b} \cdot 5^{c}$. For this number to be a square of a natural number, the exponents $a-1, b$, and $c$ must be divisible by 2.

Similarly, one-third of the number $n$ is $2^{a} \cdot 3^{b-1} \cdot 5^{c}$, and from the condition that one-third of $n$ is a cube of a natural number, we conclude that $a, b-1$, and $c$ must be divisible by 3.

One-fifth of this number, $2^{a} \cdot 3^{b} \cdot 5^{c-1}$, will be a fifth power of a natural number if $a, b$, and $c-1$ are divisible by 5.

We need to determine the smallest natural numbers $a, b$, and $c$ with these properties.

The smallest number $a$ that is divisible by 3 and 5, but not by 2, is $a=15$.

The smallest number $b$ that is divisible by 2 and 5, such that $b-1$ is divisible by 3, is $b=10$,

and the smallest number $c$ that is divisible by 2 and 3, such that $c-1$ is divisible by 5, is $c=6$.

4 points

Thus, the desired number is $n=2^{15} \cdot 3^{10} \cdot 5^{6} \quad(=30233088000000)$.

1 point

Note: In both cases, 4 points are given for three analogous conclusions/results. If the student has only one (any) of these results, 2 points should be given, and if they have two out of three results, 3 points should be given.