LCM Calculation Using Prime Factorization

The goal is to find the Least Common Multiple (LCM) of the given numbers. This requires expressing each number fully using its prime factors.

Step 1: Prime Factorize Numbers

Convert the given expressions into prime factorized forms:

• Number 1: $5^3 \times 8^2 \times 12$
  Use $8 = 2^3$ and $12 = 2^2 \times 3$.
  $5^3 \times (2^3)^2 \times (2^2 \times 3^1) = 5^3 \times 2^6 \times 2^2 \times 3^1$
  Combine powers: $2^{(6+2)} \times 3^1 \times 5^3 = 2^8 \times 3^1 \times 5^3$.
• Number 2: $5^2 \times 12^2 \times 16$
  Use $12 = 2^2 \times 3$ and $16 = 2^4$.
  $5^2 \times (2^2 \times 3)^2 \times 2^4 = 5^2 \times (2^4 \times 3^2) \times 2^4$
  Combine powers: $2^{(4+4)} \times 3^2 \times 5^2 = 2^8 \times 3^2 \times 5^2$.
• Number 3: $8^3 \times 12^2 \times 16^2$
  Use $8 = 2^3$, $12 = 2^2 \times 3$, and $16 = 2^4$.
  $(2^3)^3 \times (2^2 \times 3)^2 \times (2^4)^2 = 2^9 \times (2^4 \times 3^2) \times 2^8$
  Combine powers: $2^{(9+4+8)} \times 3^2 = 2^{21} \times 3^2$. Write as $2^{21} \times 3^2 \times 5^0$ for comparison.

Step 2: Determine Highest Powers for LCM

The prime factorizations are:

• Number 1: $2^8 \times 3^1 \times 5^3$
• Number 2: $2^8 \times 3^2 \times 5^2$
• Number 3: $2^{21} \times 3^2 \times 5^0$

Identify the maximum exponent for each prime factor (2, 3, 5):

• Max power of 2: $\max(8, 8, 21) = 21$
• Max power of 3: $\max(1, 2, 2) = 2$
• Max power of 5: $\max(3, 2, 0) = 3$

Step 3: Construct the LCM

Combine the highest powers determined:

LCM = $2^{21} \times 3^2 \times 5^3$.

Step 4: Match with Options

The result $2^{21} \times 3^2 \times 5^3$ matches Option 1.