Yes, the statement is correct! Here's a proof and explanation of the formula:

Statement:

If $k > 0$, then: $\text{gcd}(ka, kb) = k \cdot \text{gcd}(a, b)$

Proof:

The greatest common divisor (gcd) of two numbers $x$ and $y$, denoted $\text{gcd}(x, y)$, is the largest positive integer that divides both $x$ and $y$.

Let’s consider $\text{gcd}(ka, kb)$. We want to show that this equals $k \cdot \text{gcd}(a, b)$.

1. First, show that $k \cdot \text{gcd}(a, b)$ divides both $ka$ and $kb$:

   • Let $d = \text{gcd}(a, b)$. By definition, $d$ divides both $a$ and $b$.
   • Then $ka = k \cdot a$ and $kb = k \cdot b$, and since $d$ divides both $a$ and $b$, it follows that $k \cdot d$ divides both $ka$ and $kb$.
   • Therefore, $k \cdot d$ is a common divisor of $ka$ and $kb$.

2. Next, show that $k \cdot \text{gcd}(a, b)$ is the greatest common divisor:

   • Suppose $g$ is a common divisor of $ka$ and $kb$. This means $g$ divides both $ka$ and $kb$.
   • Since $g$ divides both $ka$ and $kb$, it must also divide $k$. Additionally, since $g$ divides both $ka$ and $kb$, it must divide $a$ and $b$ as well.
   • Therefore, $g$ must divide $\text{gcd}(a, b)$. Thus, $g \leq \text{gcd}(a, b)$, and so $g \leq k \cdot \text{gcd}(a, b)$.
   • From the above, we conclude that $k \cdot \text{gcd}(a, b)$ is the greatest common divisor of $ka$ and $kb$.

Conclusion:

Thus, we have shown that $\text{gcd}(ka, kb) = k \cdot \text{gcd}(a, b)$.

Would you like a more detailed breakdown or further examples of this property in action?

Related Questions:

1. How does the gcd function work for negative numbers?
2. Can we use the same formula for $k < 0$?
3. What is the relationship between the gcd and the least common multiple (LCM)?
4. How can this property be used to simplify problems involving divisibility?
5. What are some real-world applications of gcd calculations?

Tip: When working with gcd, one quick way to compute it is using the Euclidean algorithm. This method involves dividing and finding remainders, and it's very efficient for larger numbers!