Find the sum of all the common factors of 25 and 80.Multiply the result by the first common multiple of 6 and 8.What is the answer?

1. Break Down the Problem

To solve the problem, we will:

1. Find the common factors of 25 and 80.
2. Sum those common factors.
3. Find the least common multiple (LCM) of 6 and 8.
4. Multiply the sum of common factors by the LCM.

2. Relevant Concepts

• Common Factors: Factors that are shared between two numbers.
• Least Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

3. Analysis and Detail

Step 1: Find Common Factors of 25 and 80

• Factors of 25: 1, 5, 25
• Factors of 80: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
• Common Factors: The common factors of 25 and 80 are {1, 5}.

Step 2: Sum the Common Factors

$\text{Sum of Common Factors} = 1 + 5 = 6$

Step 3: Find the Least Common Multiple of 6 and 8

• The multiples of 6 are: 6, 12, 18, 24, ...
• The multiples of 8 are: 8, 16, 24, 32, ...
• The lowest common multiple (LCM) is 24.

4. Verify and Summarize

• Common factors of 25 and 80: {1, 5}.
• Sum of common factors: 6.
• LCM of 6 and 8: 24.

Final Answer

Now, multiply the sum of common factors by the LCM: $\text{Final Result} = 6 \times 24 = 144$ Thus, the answer is 144.