Understanding the Problem: Finding the LCM

The question asks us to find the Least Common Multiple (LCM) of three given numbers. These numbers are expressed using bases like 3, 8, 13, and 16, raised to certain powers. To find the LCM effectively, we need to work with the prime factorization of each number. The LCM is the smallest positive integer that is divisible by each of the given numbers.

Breaking Down the Numbers

The three numbers given are:

• Number 1: $3^3 \times 8^2 \times 13$
• Number 2: $3^2 \times 13^2 \times 16$
• Number 3: $8^3 \times 13^2 \times 16^2$

Notice that the bases 8 and 16 are not prime numbers. We need to express them in terms of their prime factors. The prime factorization of 8 is $2^3$, and the prime factorization of 16 is $2^4$.

Prime Factorization Step

Let's rewrite each number using only prime bases (2, 3, and 13). This allows us to easily compare the powers of each prime factor.

1. Number 1: $3^3 \times 8^2 \times 13 = 3^3 \times (2^3)^2 \times 13^1$ Using the power of a power rule ($(a^m)^n = a^{m \times n}$), we get: $3^3 \times 2^{(3 \times 2)} \times 13^1 = 3^3 \times 2^6 \times 13^1$ Rearranging in ascending order of bases: $2^6 \times 3^3 \times 13^1$
2. Number 2: $3^2 \times 13^2 \times 16 = 3^2 \times 13^2 \times 2^4$ Rearranging: $2^4 \times 3^2 \times 13^2$
3. Number 3: $8^3 \times 13^2 \times 16^2 = (2^3)^3 \times 13^2 \times (2^4)^2$ Applying the power of a power rule: $2^{(3 \times 3)} \times 13^2 \times 2^{(4 \times 2)} = 2^9 \times 13^2 \times 2^8$ Combining the powers of 2 using the product rule ($a^m \times a^n = a^{m+n}$): $2^{(9+8)} \times 13^2 = 2^{17} \times 13^2$ To make comparison easier, we can include $3^0$ (since $3^0 = 1$): $2^{17} \times 3^0 \times 13^2$

So, the numbers in their prime factorized form are: Number 1: $2^6 \times 3^3 \times 13^1$ Number 2: $2^4 \times 3^2 \times 13^2$ Number 3: $2^{17} \times 3^0 \times 13^2$

Calculating the LCM

To find the LCM, we take the highest power of each prime factor present across all the numbers. The prime factors involved are 2, 3, and 13.

• Highest power of 2: Comparing the powers $2^6$, $2^4$, and $2^{17}$, the highest power is $2^{17}$.
• Highest power of 3: Comparing the powers $3^3$, $3^2$, and $3^0$, the highest power is $3^3$.
• Highest power of 13: Comparing the powers $13^1$, $13^2$, and $13^2$, the highest power is $13^2$.

Therefore, the LCM is the product of these highest powers: LCM = $2^{17} \times 3^3 \times 13^2$

Comparing with Options

Now, we need to check which of the given options matches our calculated LCM. Let's examine Option 1: $3^3 \times 8^3 \times 13^2 \times 16^2$.

We convert Option 1 back into its prime factorization to compare: $3^3 \times 8^3 \times 13^2 \times 16^2$ Substitute $8 = 2^3$ and $16 = 2^4$: $= 3^3 \times (2^3)^3 \times 13^2 \times (2^4)^2$ Apply the power of a power rule: $= 3^3 \times 2^{(3 \times 3)} \times 13^2 \times 2^{(4 \times 2)}$ $= 3^3 \times 2^9 \times 13^2 \times 2^8$ Combine the powers of 2: $= 3^3 \times 2^{(9+8)} \times 13^2$ $= 3^3 \times 2^{17} \times 13^2$

This result, $2^{17} \times 3^3 \times 13^2$, exactly matches the LCM we calculated.

For completeness, let's briefly analyze the prime factorization of the other options to confirm they are incorrect:

Conclusion

By carefully converting the given numbers into their prime factorized forms and then selecting the highest power for each prime factor (2, 3, and 13), we determined the LCM to be $2^{17} \times 3^3 \times 13^2$. Option 1 is the only expression that correctly represents this LCM.