. In a company, 5 employees are doing overtime work. On the first day of the month, all the 5 employees did overtime work. Afterwards, those 5 employees do overtime work once in 3,4,6, 8 and 12 days respectively. a) On which day of the month will all the 5 employees do the overtime work together? b) Briefly explain the technique you used to solve (a).

To determine when all five employees will do overtime work together again, we need to find the smallest number of days that is a common multiple of their individual overtime schedules. This involves finding the **least common multiple (LCM)** of the numbers 3, 4, 6, 8, and 12.

### Step 1: List the periods for each employee
The periods (in days) for the employees to do overtime work are:
- Employee 1: every 3 days
- Employee 2: every 4 days
- Employee 3: every 6 days
- Employee 4: every 8 days
- Employee 5: every 12 days

### Step 2: Find the LCM of these periods
To calculate the LCM of 3, 4, 6, 8, and 12, we follow these steps:

1. **Prime factorization** of each number:
   - 3 = \(3\)
   - 4 = \(2^2\)
   - 6 = \(2 \times 3\)
   - 8 = \(2^3\)
   - 12 = \(2^2 \times 3\)

2. **Identify the highest powers** of all prime factors:
   - For 2: the highest power is \(2^3\) (from 8)
   - For 3: the highest power is \(3^1\) (from 3, 6, and 12)

3. **Calculate the LCM** by multiplying these highest powers:
   \[
   \text{LCM} = 2^3 \times 3^1 = 8 \times 3 = 24
   \]

### Step 3: Determine the day they all work together
The LCM is 24, so all five employees will do overtime together every 24 days.

### Answer:
**a)** All 5 employees will do overtime work together again on the **24th day** of the month.

**b)** The technique used to solve this problem involves calculating the **least common multiple (LCM)** of the intervals at which each employee does overtime. The LCM represents the smallest time period after which all the employees' schedules align, meaning they will all work together again on the 24th day.

Would you like more details or have any questions?

Here are some related questions you might explore:
1. What if the intervals for overtime work were different, such as 5, 7, 9, 11, and 13 days?
2. How would you find the LCM if there were more employees with different schedules?
3. Can you explain how the LCM is different from the greatest common divisor (GCD)?
4. How often will only some of the employees (but not all) work overtime together?
5. What happens if a new employee joins with a different schedule, say every 10 days?

**Tip:** The LCM is crucial for determining when multiple repeating events align. It's especially useful in planning and scheduling problems like this one.

Discover how to determine when all employees will do overtime work together using the Least Common Multiple (LCM). This problem involves finding the LCM of intervals (3, 4, 6, 8, and 12 days) at which employees work overtime. Learn the step-by-step solution and the relevance of LCM in scheduling and planning. Suitable for Grades 7-9.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation