### Instruction: The integer $m$ is between $30$ and $80$ and is a multiple of $6$. When $m$ is divided by $8,$ the remainder is $2$. Similarly, when $m$ is divided by $5,$ the remainder is $2$. What is the value of $m$?### Response: I need to find an integer $m$ that satisfies three conditions: it is a multiple of $6$, it has a remainder of $2$ when divided by $8$, and it has a remainder of $2$ when divided by $5$.
I can start by listing some multiples of $6$ that are between $30$ and $80$: $36, 42, 48, 54, 60, 66, 72, 78$.
Then I can check each of these numbers to see if they have a remainder of $2$ when divided by $8$: $36 \div 8 = 4 \text{ R } 4$, $42 \div 8 = 5 \text{ R } 2$, $48 \div 8 = 6 \text{ R } 0$, $54 \div 8 = 6 \text{ R } 6$, $60 \div 8 = 7 \text{ R } 4$, $66 \div 8 = 8 \text{ R } 2$, $72 \div 8 = 9 \text{ R } 0$, $78 \div 8 = 9 \text{ R } 6$.
So far, only $42$ and $66$ have a remainder of $2$ when divided by $8$.
Next, I can check these two numbers to see if they have a remainder of $2$ when divided by $5$: $42 \div 5 = 8 \text{ R } 2$, $66 \div 5 = 13 \text{ R } 1$.
Only $42$ has a remainder of $2$ when divided by $5$.
Therefore, $m = 42$.