Certainly! To determine the least distance that the four wheels should cover so that each wheel makes a complete number of revolutions, we need to find the least common multiple (LCM) of their circumferences. The circumferences of the wheels are 50 cm, 60 cm, 75 cm, and 100 cm. Here is a step-by-step solution:
1. List the Circumferences:
We start with the circumferences of the four wheels: 50 cm, 60 cm, 75 cm, and 100 cm.
2. Prime Factorization:
Prime factorize each circumference:
- 50 = [tex]\(2 \times 5^2\)[/tex]
- 60 = [tex]\(2^2 \times 3 \times 5\)[/tex]
- 75 = [tex]\(3 \times 5^2\)[/tex]
- 100 = [tex]\(2^2 \times 5^2\)[/tex]
3. Identify the Highest Powers of Each Prime Factor:
- For the prime number 2, the highest power is [tex]\(2^2\)[/tex] (from 60 and 100).
- For the prime number 3, the highest power is 3 (from 60 and 75).
- For the prime number 5, the highest power is [tex]\(5^2\)[/tex] (from 50, 75, and 100).
4. Calculate the LCM:
Multiply the highest powers of all prime factors together:
- LCM = [tex]\(2^2 \times 3 \times 5^2\)[/tex]
- Calculate:
[tex]\(2^2 = 4\)[/tex]
[tex]\(5^2 = 25\)[/tex]
- Multiply:
[tex]\(4 \times 3 \times 25 = 4 \times 75 = 300\)[/tex]
So, the least distance that the wheels should cover to make a complete number of revolutions is [tex]\(\textbf{300 cm}\)[/tex].
