Problem 17

Explanation

Given the conditions for integer $m$:

• $m \equiv 3 \pmod{7}$ (If divided by $7$, remainder is $3$)
• $m \equiv 2 \pmod{3}$ (If divided by $3$, remainder is $2$)

Let's check each statement:

Analysis of Statement 1

Value $m = 52$. Divided by $7$: $52 = 7 \times 7 + 3$. Remainder $3$ (Satisfied). Divided by $3$: $52 = 17 \times 3 + 1$. Remainder $1$ (Not satisfied, should be $2$). Thus, statement $1$ is incorrect.

Analysis of Statement 2

Value $m = 59$. Divided by $7$: $59 = 8 \times 7 + 3$. Remainder $3$ (Satisfied). Divided by $3$: $59 = 19 \times 3 + 2$. Remainder $2$ (Satisfied). Thus, statement $2$ is correct.

Analysis of Statement 3

Value $m = 42$. Divided by $7$: $42 = 6 \times 7 + 0$. Remainder $0$ (Not satisfied, should be $3$). Divided by $3$: $42 = 14 \times 3 + 0$. Remainder $0$ (Not satisfied, should be $2$). Thus, statement $3$ is incorrect.

Analysis of Statement 4

Value $m = 38$. Divided by $7$: $38 = 5 \times 7 + 3$. Remainder $3$ (Satisfied). Divided by $3$: $38 = 12 \times 3 + 2$. Remainder $2$ (Satisfied). Thus, statement $4$ is correct.

Since statements $2$ and $4$ are correct, the correct answer is If $(2)$ and $(4)$ are correct.