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**.