The least common multiple of $$ 3^{2} imes 11,3^{2} imes 5 $$, and $$ 2 imes 3^{2} imes 5^{2} $$ is

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The least common multiple of $$ 3^{2} 	imes 11,3^{2} 	imes 5 $$, and $$ 2 	imes 3^{2} 	imes 5^{2} $$ is

UpStudy AI · Accepted Answer

We have the following numbers with their prime factorizations:
1. $$ 3^{2} \times 11 $$
2. $$ 3^{2} \times 5 $$
3. $$ 2 \times 3^{2} \times 5^{2} $$

To find the least common multiple (LCM), we need to take the highest power of each prime factor present in any of the numbers.
- For the prime factor $$ 2 $$: The highest power is $$ 2^{1} $$ (from the third number).
- For the prime factor $$ 3 $$: The highest power is $$ 3^{2} $$ (from all three numbers).
- For the prime factor $$ 5 $$: The highest power is $$ 5^{2} $$ (from the third number).
- For the prime factor $$ 11 $$: The highest power is $$ 11^{1} $$ (from the first number).

Now, we can calculate the LCM using the highest powers of each prime factor:
$$
\text{LCM} = 2^{1} \times 3^{2} \times 5^{2} \times 11^{1}
$$
Calculate the value by following steps:
- step0: Calculate:
  $$2^{1}\times 3^{2}\times 5^{2}\times 11^{1}$$
- step1: Evaluate the power:
  $$2\times 3^{2}\times 5^{2}\times 11^{1}$$
- step2: Evaluate the power:
  $$2\times 3^{2}\times 5^{2}\times 11$$
- step3: Multiply the terms:
  $$22\times 3^{2}\times 5^{2}$$
- step4: Multiply the terms:
  $$198\times 5^{2}$$
- step5: Evaluate the power:
  $$198\times 25$$
- step6: Multiply:
  $$4950$$
The least common multiple of $$ 3^{2} \times 11, 3^{2} \times 5 $$, and $$ 2 \times 3^{2} \times 5^{2} $$ is $$ 4950 $$.

The least common multiple of , and is

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