Question

If $a$, $b$, $c$ are different integers such that $(a,b) = c$, which of the following statements is true?

• A. $c \mid a$
• B. $c \mid b$
• C. $b \mid a$
• D. None of these

Hint:

The GCD of two integers divides both of them — this is the defining property of the greatest common divisor.

Answer 1

The Correct Option is A

Step 1: Conceptual Understanding:
$(a,b) = c$ means $c$ is the greatest common divisor of $a$ and $b$.
Step 2: Explanation in Detail:
By definition of GCD, $c$ divides both $a$ and $b$. So both $\dfrac{a}{c}$ and $\dfrac{b}{c}$ are integers, i.e.\ $c \mid a$ and $c \mid b$.
Option (A) $c \mid a$ is therefore correct.
Step 3: Therefore, Stating the Final Answer
$c \mid a$ is true.