Correct Answer: B) $32$ Step 1: Prime Factorization of the given number First, we find the prime factorization of $705600$: $$705600 = 7056 \times 100$$ Since $7056 = 84^2$ and $100 = 10^2$: $$705600 = 84^2 \times 10^2 = (84 \times 10)^2 = 840^2$$ Now, prime factorize $840$: $$840 = 84 \times 10 = (2^2 \times 3 \times 7) \times (2 \times 5) = 2^3 \times 3^1 \times 5^1 \times 7^1$$ Therefore, we can write: $$705600 = (2^3 \times 3^1 \times 5^1 \times 7^1)^2 = 2^6 \times 3^2 \times 5^2 \times 7^2$$ Step 2: Condition for a factor to be a perfect square Let a generic integer be expressed by its prime factorization $N = p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$. Any factor $F$ of $N$ will be of the form $p_1^{x_1} \times p_2^{x_2} \times \dots \times p_k^{x_k}$, where $0 \le x_i \le a_i$ for each $i$. For $F$ to be a perfect square, all the exponents $x_i$ in its prime factorization must be **even numbers**. Step 3: Applying the condition to $705600$ A factor of $2^6 \times 3^2 \times 5^2 \times 7^2$ that is a perfect square will be of the form $2^a \times 3^b \times 5^c \times 7^d$, where $a, b, c, d$ must be even integers within their respective limits: * **For the prime factor $2$:** The exponent $a$ must be even and $0 \le a \le 6$. The possible values for $a$ are $\{0, 2, 4, 6\}$ ($4$ choices). * **For the prime factor $3$:** The exponent $b$ must be even and $0 \le b \le 2$. The possible values for $b$ are $\{0, 2\}$ ($2$ choices). * **For the prime factor $5$:** The exponent $c$ must be even and $0 \le c \le 2$. The possible values for $c$ are $\{0, 2\}$ ($2$ choices). * **For the prime factor $7$:** The exponent $d$ must be even and $0 \le d \le 2$. The possible values for $d$ are $\{0, 2\}$ ($2$ choices). Step 4: Calculating the total number of perfect square factors The total number of factors of $705600$ that are perfect squares is the product of the number of choices for each exponent: $$\text{Number of perfect square factors} = (\text{choices for } a) \times (\text{choices for } b) \times (\text{choices for } c) \times (\text{choices for } d)$$ $$\text{Number of perfect square factors} = 4 \times 2 \times 2 \times 2 = 32$$