To prove that $|\gcd(a, b)|$ is the greatest number under all common divisors for $a, b \in \mathbb{Z}$, we will use the following definition and properties of the greatest common divisor (gcd):

1. Definition of gcd: The greatest common divisor of two integers $a$ and $b$, denoted as $\gcd(a, b)$, is the largest positive integer $d$ such that $d$ divides both $a$ and $b$.

2. Existence and Uniqueness: For any pair of integers $a$ and $b$, there exists a unique greatest common divisor $\gcd(a, b)$, which is positive.

3. Property: If $d$ is any common divisor of $a$ and $b$, then $d \leq \gcd(a, b)$.

To prove that $|\gcd(a, b)|$ is the greatest number among all common divisors, we proceed as follows:

Step 1: gcd is a common divisor

By definition, $\gcd(a, b)$ divides both $a$ and $b$. Therefore, $\gcd(a, b)$ is a common divisor of $a$ and $b$.

Step 2: All common divisors divide the gcd

Let $d$ be any common divisor of $a$ and $b$. Then, by definition of a common divisor, $d$ divides both $a$ and $b$. This implies that there exist integers $k$ and $l$ such that: $a = dk \quad \text{and} \quad b = dl$

Since $d$ divides both $a$ and $b$, it must also divide any linear combination of $a$ and $b$. Consider the equation: $\gcd(a, b) = sa + tb$ for some integers $s$ and $t$ (this is a result from the Extended Euclidean Algorithm).

Substituting $a = dk$ and $b = dl$ into the equation, we get: $\gcd(a, b) = s(dk) + t(dl) = d(sk + tl)$

This shows that $d$ divides $\gcd(a, b)$.

Step 3: gcd is the greatest common divisor

Since $\gcd(a, b)$ is a common divisor and every common divisor $d$ of $a$ and $b$ divides $\gcd(a, b)$, it follows that $\gcd(a, b)$ is the greatest of all common divisors. More formally: $\forall \, d \text{ such that } d \mid a \text{ and } d \mid b, \, d \leq \gcd(a, b)$

Step 4: Absolute value of gcd

Since the gcd is defined to be the greatest positive integer that divides both $a$ and $b$, $|\gcd(a, b)|$ is simply $\gcd(a, b)$ if $\gcd(a, b)$ is already positive. For negative $\gcd(a, b)$, the absolute value ensures positivity, but since gcd is traditionally defined as positive, this step usually does not alter the value.

Conclusion

Thus, $|\gcd(a, b)|$ is indeed the greatest number among all common divisors of $a$ and $b$. This completes the proof.

Do you want more details or have any questions?

Here are 5 numbered questions you might ask next:

1. How does the Euclidean Algorithm work for finding the gcd of two numbers?
2. Can you show an example of using the Extended Euclidean Algorithm?
3. What is the significance of the gcd in number theory?
4. How does the gcd relate to the least common multiple (LCM)?
5. Can you prove properties of gcd for more than two integers?

Tip: When studying the properties of gcd, it is helpful to understand the Euclidean Algorithm, as it provides an efficient method for computing the gcd and offers insights into its properties.