Find the smallest number by which $5808$ should be multiplied so that the product becomes a perfect square.

### Concept & Strategy For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. To find the smallest number to multiply by, break the number down into its prime factors and identify which prime numbers have odd powers. Multiply by those exact primes to make their powers even. ### Step-by-Step Solution Find the prime factorization of $5808$. Divide by $2$ since it is even: $5808 = 2 \times 2904$ $2904 = 2 \times 1452$ $1452 = 2 \times 726$ $726 = 2 \times 363$ Now, factorize $363$. It is divisible by $3$ ($3+6+3=12$): $363 = 3 \times 121$ Recognize $121$ as a perfect square: $121 = 11 \times 11 = 11^2$ Combine all the factors: $5808 = 2^4 \times 3^1 \times 11^2$ Check the powers of each prime factor: * The power of $2$ is $4$ (Even). * The power of $11$ is $2$ (Even). * The power of $3$ is $1$ (Odd). To make the entire product a perfect square, we need the power of $3$ to be even (i.e., $3^2$). Therefore, we must multiply the number by $3$. ### Exam Strategy & Shortcut Instead of full prime factorization, you can sequentially divide by obvious squares. Divide by $4$: $5808 \div 4 = 1452$. Divide by $4$ again: $1452 \div 4 = 363$. Divide by $121$ (known square): $363 \div 121 = 3$. The remaining non-square factor is $3$. To make it a square, just multiply by $3$. ### Common Pitfall Stopping the factorization process too early or forgetting to count the total powers correctly, which can lead to mistakenly identifying a different number as the required multiplier. ### Final Answer Therefore, the correct answer is **3**.