Which of the following is the greatest measures that can be used to measure the quantities 68 m, 102 m and 119 m ?

Options

A. 34
B. 51
C. 13
D. 17

Explanation

To find the greatest measure that can be used to measure the quantities 68 m, 102 m, and 119 m, we need to determine the highest common factor (HCF) of these three numbers. Here’s how we can do it step by step:

### Step 1: Find the Prime Factorization of Each Number
- **For 68**:
- Divide by 2: \(68 ÷ 2 = 34\)
- Divide by 2 again: \(34 ÷ 2 = 17\)
- 17 is a prime number.
- So, the prime factorization of 68 is \(2^2 × 17\).

- **For 102**:
- Divide by 2: \(102 ÷ 2 = 51\)
- Divide by 3: \(51 ÷ 3 = 17\)
- 17 is a prime number.
- So, the prime factorization of 102 is \(2 × 3 × 17\).

- **For 119**:
- Divide by 7: \(119 ÷ 7 = 17\)
- 17 is a prime number.
- So, the prime factorization of 119 is \(7 × 17\).

### Step 2: Identify Common Factors
Now, we will look for the common factors in the prime factorizations:
- 68: \(2^2 × 17\)
- 102: \(2 × 3 × 17\)
- 119: \(7 × 17\)

The only common factor among all three numbers is **17**.

### Step 3: Determine the HCF
The highest common factor (HCF) is the greatest measure that can be used to measure all three quantities. Since 17 is the only common factor, the HCF is:

**HCF = 17**

### Conclusion
Thus, the greatest measure that can be used to measure the quantities 68 m, 102 m, and 119 m is **17 m**.

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