Applying the Polya's four step strategy, answer the following:

1. Bill and Sue both work a night shift. Bill has every sixth night off and Sue has every fifth night off, how many nights will it be before they are both at the same time again?

**Polya's Four-Step Problem-Solving Strategy:**

### Step 1: **Understand the Problem**
- Bill has every 6th night off, so his off nights follow the pattern: 6, 12, 18, etc.
- Sue has every 5th night off, so her off nights follow the pattern: 5, 10, 15, etc.
- We are asked to find how many nights it will take before they both have a night off on the same day.

### Step 2: **Devise a Plan**
This problem is asking for when two events coincide, so we need to find the smallest common time period. This is a typical case of finding the **least common multiple (LCM)** of the two numbers 6 and 5, because both Bill and Sue will have a night off at the same time on the least common multiple of their intervals.

### Step 3: **Carry Out the Plan**
To find the LCM of 6 and 5, we can use the following method:

- The prime factorization of 6 is \( 6 = 2 \times 3 \).
- The prime factorization of 5 is \( 5 = 5 \) (it’s already prime).

The LCM is the product of the highest powers of all prime factors involved. Since the highest powers are \( 2^1 \), \( 3^1 \), and \( 5^1 \), we have:

\[
\text{LCM}(6, 5) = 2^1 \times 3^1 \times 5^1 = 30.
\]

### Step 4: **Look Back and Check**
- Bill has every 6th night off: 6, 12, 18, 24, 30...
- Sue has every 5th night off: 5, 10, 15, 20, 25, 30...
- They both have a night off together on the 30th night.

Thus, Bill and Sue will both have a night off at the same time every 30 nights.

### Final Answer:
It will be **30 nights** before Bill and Sue both have a night off at the same time.

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Would you like further clarification on any step or need additional details?

Here are some related questions you might find useful:

1. What is the definition of least common multiple (LCM)?
2. How would you find the LCM of three numbers?
3. How do prime factorization and LCM relate?
4. How would the answer change if Bill had every 8th night off?
5. Can LCM help solve problems in scheduling outside this example?

**Tip**: Always break down problems involving recurring events into their underlying numerical patterns!

Bill and Sue both work a night shift. Bill has every sixth night off and Sue has every fifth night off, how many nights will it be before they are both at the same time again?

Discover how to find the least common multiple (LCM) of 6 and 5 to determine when Bill and Sue will both have the same night off again. This problem explores number theory concepts and prime factorization, suitable for students in Grades 6-8.

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