Step 1: Find Prime Factors of 1083

Check divisibility by prime numbers. The sum of digits of $1083$ is $1 + 0 + 8 + 3 = 12$, which is divisible by $3$. Dividing $1083$ by $3$ gives $361$.

$1083 = 3 \times 361$

Step 2: Factorize 361

Recognize that $361$ is the square of $19$ ($19 \times 19 = 361$).

$361 = 19^2$

Step 3: Combine Prime Factors

Write $1083$ in terms of its prime factorization with exponents: $3^1 \times 19^2$.

$1083 = 3^1 \times 19^2$

Step 4: Determine the Smallest Multiplier

For a number to be a perfect square, all exponents in its prime factorization must be even. Here, the exponent of $3$ is $1$ (odd). Multiplying by $3$ makes it $3^2$, resulting in a perfect square.

$1083 \times 3 = 3^2 \times 19^2 = (3 \times 19)^2 = 57^2 = 3249$