Three measuring rods are of length 120 cm, 100 cm and 150 cm. Find the least length of the fence that can be measure an exact number of times, using any of the rods. How many times each rod will be used to measure the length of the fence?

Three measuring rods are of length 120 cm, 100 cm and 150 cm. Find the least length of the fence that can be measure an exact number of times, using any of the rods. How many times each rod will be used to measure the length of the fence?

To find the least length of the fence that can be measured exactly using any of the given rods, we need to determine the **Least Common Multiple (LCM)** of the rod lengths.

Given rod lengths:
$ \displaystyle 120 $ cm, $ \displaystyle 100 $ cm, $ \displaystyle 150 $ cm

### Step 1: Find the LCM
Prime factorizations:

$ \displaystyle 120 = 2^3 \times 3 \times 5 $
$ \displaystyle 100 = 2^2 \times 5^2 $
$ \displaystyle 150 = 2 \times 3 \times 5^2 $

The LCM is obtained by taking the highest power of each prime factor:

$ \displaystyle LCM = 2^3 \times 3 \times 5^2 $

$ \displaystyle = 8 \times 3 \times 25 $

$ \displaystyle = 600 $ cm

Thus, the least length of the fence that can be measured exactly using any of the rods is **600 cm**.

### Step 2: Number of times each rod is used
- Using the $ \displaystyle 120 $ cm rod: $ \displaystyle \frac{600}{120} = 5 $ times
- Using the $ \displaystyle 100 $ cm rod: $ \displaystyle \frac{600}{100} = 6 $ times
- Using the $ \displaystyle 150 $ cm rod: $ \displaystyle \frac{600}{150} = 4 $ times

### Final Answer:
The least length of the fence is **600 cm**.
- The $ \displaystyle 120 $ cm rod is used **5 times**.
- The $ \displaystyle 100 $ cm rod is used **6 times**.
- The $ \displaystyle 150 $ cm rod is used **4 times**.

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