We are given that:

$360 = 2^3 \times 3^2 \times 5$

To form a perfect square, all prime factors must have even exponents. Currently, the exponents of the factors are:

• $2^3$ (odd exponent)
• $3^2$ (even exponent)
• $5^1$ (odd exponent)

To make all exponents even, we need to multiply by $2^1$ (to make $2^3$ become $2^4$) and $5^1$ (to make $5^1$ become $5^2$).

Thus, the smallest number to multiply by is:

$2 \times 5 = 10$

So, the answer is 10.

Would you like a further explanation? Here are some related questions:

1. What is a perfect square, and why do all prime factors need even exponents?
2. How do you find the prime factorization of a number systematically?
3. If a number is already a perfect square, how can you verify it using its prime factorization?
4. What is the smallest number that 180 can be multiplied by to make it a square?
5. How does this method apply to finding the least common multiple (LCM) of numbers?

Tip: A number is a perfect square if and only if all the exponents in its prime factorization are even.