Amanita, let's solve the problem step by step! The question asks us to find the Least Common Multiple (LCM) of each pair of numbers using their prime factorization. Let's break down the process for each pair:

Step-by-Step Plan

1. Prime Factorization: Write each number as a product of its prime factors.
2. Find the LCM: For each pair, the LCM is found by taking the highest power of each prime that appears in either number.
3. Calculate the LCM: Multiply these highest powers together to get the LCM for each pair.

Let's solve each part one by one.

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(1) $3 \times 5$ and $2 \times 5^2$

• First number: $3 \times 5$
• Second number: $2 \times 5^2$

Let's find the LCM.

Amanita, let's look at the first pair:

(1) $3 \times 5$ and $2 \times 5^2$

• $3 \times 5 = 15$
• $2 \times 5^2 = 50$

To find the Least Common Multiple (LCM), we use the highest power of each prime number from both numbers:

• For 2: highest power is $2^1$
• For 3: highest power is $3^1$
• For 5: highest power is $5^2$

So, the LCM is:

$2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150$

The LCM of 15 and 50 is 150.

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Let's move to the next pair.

(2) $2^3 \times 3$ and $2 \times 3^2 \times 5$

• First number: $2^3 \times 3$
• Second number: $2 \times 3^2 \times 5$

Let's find the LCM. For the second pair, Amanita:

(2) $2^3 \times 3$ and $2 \times 3^2 \times 5$

• $2^3 \times 3 = 24$
• $2 \times 3^2 \times 5 = 90$

To find the LCM, we take the highest power of each prime:

• For 2: highest power is $2^3$
• For 3: highest power is $3^2$
• For 5: highest power is $5^1$

So, the LCM is:

$2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$

The LCM of 24 and 90 is 360.

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Let's continue to the next pair.

(3) 14 and 63

Let's find their prime factorizations and then the LCM. For the third pair, Amanita:

(3) 14 and 63

• 14 can be written as $2 \times 7$
• 63 can be written as $3^2 \times 7$

To find the LCM, we take the highest power of each prime:

• For 2: $2^1$
• For 3: $3^2$
• For 7: $7^1$

So, the LCM is:

$2^1 \times 3^2 \times 7^1 = 2 \times 9 \times 7 = 126$

The LCM of 14 and 63 is 126.

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Now, let's solve the last pair.

(4) 30 and 56

Let's find their prime factorizations and then the LCM. For the last pair, Amanita:

(4) 30 and 56

• 30 can be written as $2 \times 3 \times 5$
• 56 can be written as $2^3 \times 7$

To find the LCM, we take the highest power of each prime:

• For 2: $2^3$
• For 3: $3^1$
• For 5: $5^1$
• For 7: $7^1$

So, the LCM is:

$2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 = 840$

The LCM of 30 and 56 is 840.

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Summary of Answers

1. LCM of $3 \times 5$ and $2 \times 5^2$ is 150.
2. LCM of $2^3 \times 3$ and $2 \times 3^2 \times 5$ is 360.
3. LCM of 14 and 63 is 126.
4. LCM of 30 and 56 is 840.

If you have any questions about how we found these, let me know!