let a,b,c and d be positive integers and let gcd(a,b)=d. Prove that if c|a and c|b, then c|d/

To prove: If \( c \mid a \) and \( c \mid b \), then \( c \mid d \), where \( d = \gcd(a, b) \).

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### **Proof:**

#### 1. Definition of \( \gcd(a, b) \):
By definition, the greatest common divisor \( d \) of \( a \) and \( b \) satisfies:
\[
d = \gcd(a, b) = \text{largest integer that divides both } a \text{ and } b.
\]
Thus:
\[
d \mid a \quad \text{and} \quad d \mid b.
\]

#### 2. Assumptions:
It is given that:
\[
c \mid a \quad \text{and} \quad c \mid b.
\]
This means there exist integers \( k_1 \) and \( k_2 \) such that:
\[
a = k_1 c \quad \text{and} \quad b = k_2 c.
\]

#### 3. Expressing \( d \) as a linear combination:
The \( \gcd(a, b) \) can always be written as a linear combination of \( a \) and \( b \):
\[
d = p a + q b,
\]
for some integers \( p \) and \( q \).

Substituting \( a = k_1 c \) and \( b = k_2 c \) into this equation:
\[
d = p (k_1 c) + q (k_2 c).
\]

Factoring \( c \) out:
\[
d = c (p k_1 + q k_2).
\]

#### 4. Conclusion:
Since \( p k_1 + q k_2 \) is an integer, we have:
\[
c \mid d.
\]

Thus, \( c \mid d \), as required.

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### **Key Points in the Proof:**
1. \( d \) divides both \( a \) and \( b \), so it can be expressed in terms of them.
2. \( c \) dividing \( a \) and \( b \) implies that \( c \) divides any integer linear combination of \( a \) and \( b \).
3. Using the linear combination property of the \( \gcd \), we conclude that \( c \mid d \).

---

Do you want further clarification or a numerical example? 

### **Questions to Deepen Understanding:**
1. What are some examples of \( a, b, c, \) and \( d \) that satisfy this property?
2. Can you extend this proof to more than two integers (e.g., \( \gcd(a, b, c) \))?
3. Why is the linear combination property of \( \gcd \) critical in this proof?
4. How does this proof depend on \( c \) being a divisor of both \( a \) and \( b \)?
5. What happens if \( c \) does not divide \( a \) or \( b \)?

### **Tip:**
To test divisibility-related proofs, always try substituting numerical examples to verify each logical step.

Let a, b, c, and d be positive integers and let gcd(a, b) = d. Prove that if c | a and c | b, then c | d.

Explore the proof that if c divides both a and b, then c also divides gcd(a, b). This problem covers key number theory concepts such as divisibility, the definition of GCD, and the linear combination property of GCD, with a detailed step-by-step explanation.

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