3. A positive integer $m$ has the property that when multiplied by 12 , the result is a four-digit number $n$ of the form $20 A 2$ for some digit $A$. What is the 4 digit number, $n ?$
Solution 1: For a number to be divisible by 3 , the sum of the digits must be a multiple of 3 , so $3 \mid(A+4)$, which means $A \in\{2,5,8\}$.
For a number to be divisible by 4 , the number formed by the last two digits must be divisible by 4 , so $4 \mid(10 A+2)$. This gives $A \in\{1,3,5,7,9\}$.
The only number in common is $A=5$, so $n=2052$.
Solution 2: The number $n-12$ is divisible by 5 , so $n$ is of the form $60 k+12$ for some $k$. $60 \times 34+12=2052$, so $n=2052$. It is easy to verify that no other value of $k$ works.