4. Let $a, b$, and $c$ be natural numbers such that $a<b$ and the following conditions hold: $D(a, b)=4, D(a, b, c)=2$ and $V(a, b, c)=400$. Determine all triples $(a, b, c)$ of natural numbers that satisfy the given conditions.
First solution.

Since $V(a, b, c)=400$, we factorize the number 400 into prime factors:

$400=2 \cdot 2 \cdot 2 \cdot 2 \cdot 5 \cdot 5=2^{4} \cdot 5^{2}$.

1 POINT

From the condition $D(a, b)=4$, we conclude that the numbers $a$ and $b$ are multiples of the number 4.

From the factorization of the number 400 into prime factors, we conclude that $a, b \in\{4,16,20,80,100,400\}$. 2 POINTS

From the factorization of the number 400 into prime factors and the condition $D(a, b, c)=2$, we conclude that the number $c$ is a multiple of the number 2, but not of the number 4, so $c \in\{2,10,50\}$.

2 POINTS

If $c=2$, then $a, b \in\{4,400\}$ and $a, b \in\{16,100\}$.

1 POINT

If $c=10$, then $a, b \in\{4,400\}$ and $a, b \in\{16,100\}$.

1 POINT

If $c=50$, then $a, b \in\{4,16\}, a, b \in\{4,80\}, a, b \in\{4,400\}, a, b \in\{16,20\}, a, b \in\{16,100\}$.

2 POINTS

The sought ordered triples of numbers are:

$(4,400,2),(16,100,2),(4,400,10),(16,100,10)$,

$(4,16,50),(4,80,50),(4,400,50),(16,20,50),(16,100,50)$.

1 POINT

Or, the student can present the sought triples of numbers in a table.

| $c$ | 2 | 2 | 10 | 10 | 50 | 50 | 50 | 50 | 50 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| $a$ | 4 | 16 | 4 | 16 | 4 | 4 | 4 | 16 | 16 |
| $b$ | 400 | 100 | 400 | 100 | 16 | 80 | 400 | 20 | 100 |

TOTAL 10 POINTS

Note: If the student does not have the prime factorization, but has all the solutions with appropriate justifications, they should receive a maximum of 10 points.