How many natural numbers from 1 to 900 are not multiples of any of the numbers 2, 3 and 5?

Options

A. 160
B. 240
C. 320
D. 280

Answer

To find how many natural numbers from 1 to 900 are not multiples of any of the numbers 2, 3, and 5, we can use the principle of Inclusion-Exclusion.

### Step-by-Step Solution:

1. **Calculate the total natural numbers from 1 to 900**:
- The total count of natural numbers from 1 to 900 is simply 900.

2. **Calculate the number of multiples of each individual number (2, 3, and 5)**:
- **Multiples of 2**:
\[ \text{Count} = \left\lfloor \frac{900}{2} \right\rfloor = 450 \]
- **Multiples of 3**:
\[ \text{Count} = \left\lfloor \frac{900}{3} \right\rfloor = 300 \]
- **Multiples of 5**:
\[ \text{Count} = \left\lfloor \frac{900}{5} \right\rfloor = 180 \]

3. **Calculate the number of multiples of the least common multiples (LCM) of pairs of numbers**:
- **Multiples of 6 (LCM of 2 and 3)**:
\[ \text{Count} = \left\lfloor \frac{900}{6} \right\rfloor = 150 \]
- **Multiples of 10 (LCM of 2 and 5)**:
\[ \text{Count} = \left\lfloor \frac{900}{10} \right\rfloor = 90 \]
- **Multiples of 15 (LCM of 3 and 5)**:
\[ \text{Count} = \left\lfloor \frac{900}{15} \right\rfloor = 60 \]

4. **Calculate the number of multiples of the LCM of all three numbers (2, 3, and 5)**:
- **Multiples of 30 (LCM of 2, 3, and 5)**:
\[ \text{Count} = \left\lfloor \frac{900}{30} \right\rfloor = 30 \]

5. **Apply the Inclusion-Exclusion Principle**:
- The total number of multiples of 2, 3, or 5 is given by:
\[ N(2 \cup 3 \cup 5) = N(2) + N(3) + N(5) - N(2 \cap 3) - N(2 \cap 5) - N(3 \cap 5) + N(2 \cap 3 \cap 5) \]
- Substituting the values:
\[ N(2 \cup 3 \cup 5) = 450 + 300 + 180 - 150 - 90 - 60 + 30 \]
- Simplifying this gives:
\[ N(2 \cup 3 \cup 5) = 450 + 300 + 180 - 150 - 90 - 60 + 30 = 660 \]

6. **Calculate the number of natural numbers not multiples of 2, 3, or 5**:
- The required count is:
\[ \text{Natural numbers not multiples of 2, 3, or 5} = 900 - N(2 \cup 3 \cup 5) = 900 - 660 = 240 \]

### Final Answer:
The number of natural numbers from 1 to 900 that are not multiples of 2, 3, or 5 is **240**.