Question

question 5 points 3 john, amy, and stephanie all work at the movie theater. john works every 3 days, amy works every 2 days, stephanie works every 6 days. if they work all together today, how many days will pass before they all work together again? 12 6 18 9

Explanation:

Step1: Identify the problem type

We need to find the least common multiple (LCM) of the numbers 3, 2, and 6, as we want to determine when John (every 3 days), Amy (every 2 days), and Stephanie (every 6 days) will all work together again.

Step2: List the multiples

• Multiples of 3: 3, 6, 9, 12, 15, 18, ...
• Multiples of 2: 2, 4, 6, 8, 10, 12, ...
• Multiples of 6: 6, 12, 18, 24, ...

Step3: Find the LCM

The smallest number that appears in all three lists of multiples is 6. Wait, no, wait. Wait, 6 is a multiple of 2 and 3. Wait, let's check again. Wait, 3, 2, 6. Let's use prime factorization.

• Prime factors of 3: \( 3 = 3 \)
• Prime factors of 2: \( 2 = 2 \)
• Prime factors of 6: \( 6 = 2 \times 3 \)

The LCM is the product of the highest powers of all prime factors involved. So for prime 2, the highest power is \( 2^1 \), for prime 3, the highest power is \( 3^1 \). So LCM \( = 2 \times 3 = 6 \)? Wait, no, wait. Wait, Stephanie works every 6 days, Amy every 2 days, John every 3 days. Let's check when they all meet. Let's list the days each works:

• John: 3, 6, 9, 12, ...
• Amy: 2, 4, 6, 8, ...
• Stephanie: 6, 12, 18, ...

So the first common day is 6? Wait, but wait, the options include 6. Wait, but let's check again. Wait, maybe I made a mistake. Wait, 3, 2, 6. The LCM of 3, 2, and 6. Let's calculate LCM(3,2,6). Since 6 is a multiple of both 2 and 3, the LCM of 3, 2, and 6 is 6. Wait, but let's confirm. Let's see:

John works on day 3, 6, 9, 12...

Amy works on day 2, 4, 6, 8...

Stephanie works on day 6, 12, 18...

So the first day they all work together again is day 6. So the number of days that pass is 6.

Wait, but let's check the options. The options are 12, 6, 18, 9. So 6 is an option. So the answer should be 6.

Wait, but let's do prime factorization properly.

Prime factors:

• 2: prime factor is 2
• 3: prime factor is 3
• 6: prime factors are 2 and 3

So LCM is the product of the highest powers of all primes present. So 2^1 * 3^1 = 6. So LCM(2,3,6) = 6.

Answer:

6 (corresponding to the option with 6)