What is the smallest possible length that can be exactly measured by the scales of lengths 3 cm, 5 cm and 10 cm?

Options

A. 15 cm
B. 30 cm
C. 28 cm
D. 40 cm

Answer

To find the smallest possible length that can be exactly measured by scales of lengths 3 cm, 5 cm, and 10 cm, we need to calculate the Least Common Multiple (LCM) of these three lengths. Here's a step-by-step solution:

### Step 1: Identify the lengths
We have three lengths:
- L1 = 3 cm
- L2 = 5 cm
- L3 = 10 cm

### Step 2: Find the prime factorization of each length
- The prime factorization of 3 is \(3^1\).
- The prime factorization of 5 is \(5^1\).
- The prime factorization of 10 is \(2^1 \times 5^1\).

### Step 3: Determine the highest power of each prime factor
To find the LCM, we take the highest power of each prime factor from the factorizations:
- For the prime number 2: the highest power is \(2^1\) (from 10).
- For the prime number 3: the highest power is \(3^1\) (from 3).
- For the prime number 5: the highest power is \(5^1\) (from both 5 and 10).

### Step 4: Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
\[ \text{LCM} = 2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 \]
Calculating this gives:
\[ 2 \times 3 = 6 \]
\[ 6 \times 5 = 30 \]

### Step 5: Conclusion
Thus, the smallest possible length that can be exactly measured by the scales of lengths 3 cm, 5 cm, and 10 cm is **30 cm**.

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