Introduction:
To find the smallest number by which 10584 should be multiplied in order to get a perfect square, we need to analyze the prime factorization of 10584.

Prime Factorization of 10584:
To find the prime factorization of 10584, we divide it by prime numbers starting from 2 until we can no longer divide it evenly.

10584 ÷ 2 = 5292
5292 ÷ 2 = 2646
2646 ÷ 2 = 1323
1323 ÷ 3 = 441
441 ÷ 3 = 147
147 ÷ 3 = 49
49 ÷ 7 = 7
7 ÷ 7 = 1

The prime factorization of 10584 is 2^3 × 3 × 7^2.

Perfect Square:
A perfect square is a number that can be expressed as the product of an integer multiplied by itself.

To make a perfect square, every prime factor must have an even exponent. In other words, all the exponents of the prime factors must be divisible by 2.

Determining the Smallest Number:
To find the smallest number by which 10584 should be multiplied to get a perfect square, we need to determine the smallest power of each prime factor that makes the exponent even.

Prime factorization of 10584:
2^3 × 3 × 7^2

To make the exponents of 2, 3, and 7 even, we need to multiply 10584 by the following numbers:

2^1 to make 2^3 even
3^1 to make 3^1 even (already even)
7^2 to make 7^2 even

Therefore, the smallest number by which 10584 should be multiplied to get a perfect square is:

2^1 × 3^1 × 7^2 = 2 × 3 × 7^2 = 294.

Conclusion:
To make 10584 a perfect square, it should be multiplied by the smallest number, which is 294.