1. **Problem Statement:** We are given a positive integer $n$ and three statements about divisibility involving $n$ and $n^2$. We need to determine which statements are true. 2. **Recall some divisibility rules:** - If $a$ divides $b$, we write $a \mid b$. - If $n$ is a multiple of $k$, then $k \mid n$. - For prime factorization, if $n^2$ is divisible by a number, the prime factors must appear with at least twice the exponent in $n$. 3. **Analyze each statement:** **Statement 1:** If $n$ is a multiple of 8, then $n^2$ is a multiple of 4. - Since $8 \mid n$, $n = 8m$ for some integer $m$. - Then $n^2 = (8m)^2 = 64m^2$. - Since $64m^2$ is divisible by 4, $4 \mid n^2$. - **Statement 1 is true.** **Statement 2:** If $n^2$ is a multiple of 18, then $n$ is a multiple of 9. - Prime factorize 18: $18 = 2 \times 3^2$. - If $18 \mid n^2$, then $2 \mid n^2$ and $3^2 \mid n^2$. - For $2 \mid n^2$, $2 \mid n$ (since 2 is prime). - For $3^2 \mid n^2$, $3 \mid n$ (since exponents double in $n^2$). - But does $9 \mid n$ necessarily follow? - Consider $n = 6$: $n^2 = 36$, which is divisible by 18, but $6$ is not divisible by 9. - So **Statement 2 is false.** **Statement 3:** If $n^2$ is a multiple of 18, then $n$ is a multiple of 2. - From above, $18 \mid n^2$ implies $2 \mid n^2$. - Since 2 is prime, $2 \mid n$. - So **Statement 3 is true.** 4. **Final conclusion:** Statements 1 and 3 are true; Statement 2 is false.