Runners Meeting Time Calculation

This problem involves two runners moving on a circular path at constant speeds. We are given the time each runner takes to complete one full round.

• Runner 1 takes 6 minutes per round.
• Runner 2 takes 8 minutes per round.

They start together at the start line. We want to find the time when they meet again for the first time specifically at the start line.

For both runners to be at the start line at the same time, the total time elapsed must be a multiple of the time each runner takes to complete one round. The first runner will be at the start line after 6 minutes, 12 minutes, 18 minutes, 24 minutes, and so on (multiples of 6). The second runner will be at the start line after 8 minutes, 16 minutes, 24 minutes, 32 minutes, and so on (multiples of 8).

To find the first time they meet again at the start line, we need to find the smallest time that is a multiple of both 6 and 8. This is known as the Least Common Multiple (LCM) of 6 and 8.

Finding the LCM of 6 and 8

We can find the LCM by listing multiples or by using prime factorization.

Using Prime Factorization:

First, find the prime factorization of each number:

• Prime factors of 6: $6 = 2 \times 3$
• Prime factors of 8: $8 = 2 \times 2 \times 2 = 2^3$

To find the LCM, we take the highest power of all unique prime factors present in the factorizations.

• Unique prime factors are 2 and 3.
• Highest power of 2 is $2^3$ (from the factorization of 8).
• Highest power of 3 is $3^1$ (from the factorization of 6).

LCM(6, 8) = $2^3 \times 3 = 8 \times 3 = 24$

Using Listing Multiples:

List the multiples of each number until a common multiple is found:

• Multiples of 6: 6, 12, 18, 24, 30, ...
• Multiples of 8: 8, 16, 24, 32, ...

The smallest common multiple is 24.

Therefore, the runners will meet again for the first time at the start line after 24 minutes.