What is the least common multiple (LCM) of the numbers 90, 60, 75 and 35?

Question

Accepted Answer

To determine the least common multiple (LCM) of the numbers 90, 60, 75, and 35, we need to find the smallest number that is a multiple of all these numbers. Let's break down the process:

1. **Prime Factorization**:
   - 90 = 2 × 3^2 × 5
   - 60 = 2^2 × 3 × 5
   - 75 = 3 × 5^2
   - 35 = 5 × 7

2. **Identify the Highest Power of Each Prime**:
   - For the prime number 2, the highest power is 2^2 (from 60).
   - For the prime number 3, the highest power is 3^2 (from 90).
   - For the prime number 5, the highest power is 5^2 (from 75).
   - For the prime number 7, the highest power is 7 (from 35).

3. **Calculate the LCM**:
   - LCM = 2^2 × 3^2 × 5^2 × 7
   - LCM = 4 × 9 × 25 × 7
   - LCM = 6300

Thus, the least common multiple of 90, 60, 75, and 35 is 6300, which corresponds to Option 2.

**Explanation for Incorrect Options**:

- **Option 1 (2700)**: This number is not a multiple of 35, as 2700 divided by 35 does not result in an integer. Therefore, it cannot be the LCM of the given numbers.

- **Option 3 (4250)**: This number is not a multiple of 60, as 4250 divided by 60 does not result in an integer. Therefore, it cannot be the LCM of the given numbers.

- **Option 4 (2750)**: This number is not a multiple of 90, as 2750 divided by 90 does not result in an integer. Therefore, it cannot be the LCM of the given numbers.

In conclusion, the correct answer is Option 2 (6300), as it is the smallest number that is a multiple of all the given numbers.