Online Course Discussion Forum
Math Challenge I-C Spring 2019 Lecture 2 Assignment
Since the question was not available on the online assignment, it was not considered for grading.
Remember all homework questions, as well as the in-class examples are available in the class book.
Here is the missing question:
Suppose $A,B,C$ are integers $\geq 2$ with (i) $\gcd(A,B) = 12$, (ii) $\text{lcm}(A,B) = 396$, and (iii) $\gcd(B,C) = 33$.
What are the possibilities for $\gcd(A,C)$?
After you've tried it, see the next post for the solution.
And here is the solution:
$396 = 2^2\cdot 3^2\cdot 11$, so one of $A$ or $B$ is divisible by $11$. Since $\gcd(B,C) = 33 = 3\cdots 11$, it must be the case that $B$ is divisible by $11$.
This implies that $A$ is not divisible by $11$. Further, since $\gcd(A,B) = 12 = 2^2\cdot 3$ and $\gcd(B,C) = 3\cdot 11$, we know that $C$ is not divisible by $2$. Hence, $\gcd(A,C)$ is a power of $3$, so either $3$ or $9$ (both of which are possible).
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