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 $m$?
According to the problem statement, we have the system of linear congruences \begin{align*}
m &\equiv 0 \pmod{6} \\
m &\equiv 2 \pmod{8} \\
m &\equiv 2 \pmod{5}.
\end{align*} It follows by the Chinese Remainder Theorem that $m \equiv 2 \pmod{40}$. The only number that satisfies this criterion for $30 \le m \le 80$ is $m = \boxed{42}$, which is indeed divisible by $6$.
42
Level 1
Number Theory
Problem: 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 $m$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the integer $m$, we start by identifying the multiples of $6$ that lie between $30$ and $80$. The multiples of $6$ in this range are $30, 36, 42, 48, 54, 60, 66, 72, 78$. 

Next, we apply the condition that when $m$ is divided by $8$, the remainder is $2$. This can be expressed mathematically as $m \equiv 2 \pmod{8}$. We check each multiple of $6$:

- $30 \div 8 = 3$ remainder $6$ (not valid)
- $36 \div 8 = 4$ remainder $4$ (not valid)
- $42 \div 8 = 5$ remainder $2$ (valid)
- $48 \div 8 = 6$ remainder $0$ (not valid)
- $54 \div 8 = 6$ remainder $6$ (not valid)
- $60 \div 8 = 7$ remainder $4$ (not valid)
- $66 \div 8 = 8$ remainder $2$ (valid)
- $72 \div 8 = 9$ remainder $0$ (not valid)
- $78 \div 8 = 9$ remainder $6$ (not valid)

From this check, the valid candidates are $42$ and $66$. 

Next, we apply the second condition that when $m$ is divided by $5$, the remainder is also $2$. This can be expressed as $m \equiv 2 \pmod{5}$. We check our valid candidates:

- For $42$: $42 \div 5 = 8$ remainder $2$ (valid)
- For $66$: $66 \div 5 = 13$ remainder $1$ (not valid)

Since $66$ does not satisfy the second condition, we conclude that the only value of $m$ that meets all the criteria is $42$. 

Thus, the final answer is $42$.<|END_OF_LATENT|> The final answer is $42$.