How many multiples of both 3 and 5 are there from numbers 1 to 150?

Options

A. 8
B. 10
C. 11
D. 30

Answer

To find how many multiples of both 3 and 5 exist between the numbers 1 and 150, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of 3 and 5
The multiples of both 3 and 5 will be the multiples of their LCM.
- LCM of 3 and 5 is calculated as:
\[ \text{LCM}(3, 5) = 3 \times 5 = 15 \]

### Step 2: Identify the Range
We need to find the multiples of 15 from 1 to 150.

### Step 3: Determine the First and Last Multiple of 15 in the Range
- The first multiple of 15 is 15 itself.
- The last multiple of 15 within 150 can be found by dividing 150 by 15:
\[ \frac{150}{15} = 10 \]
Thus, the last multiple is:
\[ 15 \times 10 = 150 \]

### Step 4: List the Multiples of 15
The multiples of 15 from 1 to 150 are:
- 15, 30, 45, 60, 75, 90, 105, 120, 135, 150

### Step 5: Count the Multiples
To count the multiples, we can see that they are:
- 15 × 1 = 15
- 15 × 2 = 30
- 15 × 3 = 45
- 15 × 4 = 60
- 15 × 5 = 75
- 15 × 6 = 90
- 15 × 7 = 105
- 15 × 8 = 120
- 15 × 9 = 135
- 15 × 10 = 150

Counting these, we find there are a total of 10 multiples of 15.

### Final Answer
Thus, the number of multiples of both 3 and 5 from numbers 1 to 150 is **10**.

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