The LCM of 5³ × 8² × 12, 5² × 12² × 16 and 8³ × 12² × 16² is:

The correct option is 2²¹ × 3² × 5³.

To find the Least Common Multiple (LCM) of the given numbers, we must first express each number in its prime factored form (using base prime numbers such as 2, 3, and 5).

Step 1: Analyze and prime factorize the first expression
The first expression is:
$$5^3\times 8^2\times 12$$
We can express the composite bases 8 and 12 in terms of their prime factors:
$$8=2^3\Rightarrow 8^2={\left(2^3\right)}^2=2^6$$
$$12=2^2\times 3$$
Substituting these back into the first expression:
$$5^3\times 2^6\times 2^2\times 3=2^{6+2}\times 3\times 5^3=2^8\times 3\times 5^3$$

Step 2: Analyze and prime factorize the second expression
The second expression is:
$$5^2\times {12}^2\times 16$$
We factorize 12 and 16 into their prime bases:
$${12}^2={(2^2\times 3)}^2=2^4\times 3^2$$
$$16=2^4$$
Substituting these back into the second expression:
$$5^2\times 2^4\times 3^2\times 2^4=2^{4+4}\times 3^2\times 5^2=2^8\times 3^2\times 5^2$$

Step 3: Analyze and prime factorize the third expression
The third expression is:
$$8^3\times {12}^2\times {16}^2$$
We factorize 8, 12, and 16 into their prime bases:
$$8^3={\left(2^3\right)}^3=2^9$$
$${12}^2=2^4\times 3^2$$
$${16}^2={\left(2^4\right)}^2=2^8$$
Substituting these back into the third expression:
$$2^9\times 2^4\times 3^2\times 2^8=2^{9+4+8}\times 3^2=2^{21}\times 3^2$$

Step 4: Determine the LCM
To find the LCM, we take the highest power of each prime factor that appears across the three simplified expressions:
1. First Expression: $2^8\times 3^1\times 5^3$
2. Second Expression: $2^8\times 3^2\times 5^2$
3. Third Expression: $2^{21}\times 3^2\times 5^0$

Comparing the exponents of each prime:
- For base 2: The powers are 8, 8, and 21. The highest power is 21.
- For base 3: The powers are 1, 2, and 2. The highest power is 2.
- For base 5: The powers are 3, 2, and 0. The highest power is 3.

Therefore, the LCM is:
$$2^{21}\times 3^2\times 5^3$$
This perfectly matches the correct option.