The pieces of timber 42m, 49 m and 63 m long have to be divided into planks of the same lengthd. What is the greated possible length of each plank ? How many stracks will be there ?

Explanation

To solve the problem of dividing the pieces of timber of lengths 42 m, 49 m, and 63 m into planks of the same maximum length, we need to find the greatest common divisor (GCD) of the three lengths. Here’s how to do it step by step:

### Step 1: Find the Prime Factorization of Each Length
1. **For 42 m:**
- Divide by 2: \( 42 \div 2 = 21 \)
- Divide by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 42 is \( 2 \times 3 \times 7 \).

2. **For 49 m:**
- Divide by 7: \( 49 \div 7 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 49 is \( 7 \times 7 \) or \( 7^2 \).

3. **For 63 m:**
- Divide by 3: \( 63 \div 3 = 21 \)
- Divide by 3 again: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 63 is \( 3^2 \times 7 \).

### Step 2: Identify the Common Factors
Now, we look for the common prime factors in the factorizations:
- 42: \( 2 \times 3 \times 7 \)
- 49: \( 7^2 \)
- 63: \( 3^2 \times 7 \)

The only common prime factor among all three numbers is \( 7 \).

### Step 3: Determine the Greatest Common Divisor (GCD)
The GCD is the product of the lowest powers of all common prime factors:
- The GCD of 42, 49, and 63 is \( 7^1 = 7 \).

### Step 4: Calculate the Number of Planks
Now that we have the maximum length of each plank (7 m), we can find out how many planks can be made from each piece of timber:
1. **For 42 m:**
- Number of planks = \( \frac{42}{7} = 6 \)

2. **For 49 m:**
- Number of planks = \( \frac{49}{7} = 7 \)

3. **For 63 m:**
- Number of planks = \( \frac{63}{7} = 9 \)

### Step 5: Total Number of Planks
Now, we add the number of planks from each length:
- Total planks = \( 6 + 7 + 9 = 22 \)

### Final Answer
- The greatest possible length of each plank is **7 meters**.
- The total number of planks (or stacks) is **22**.