To solve the problem of finding how many multiples of 3 there are from 1 to 100 that are not multiples of 2, we can follow these steps:
### Step 1: Identify the multiples of 3 from 1 to 100
The multiples of 3 within this range can be found by listing them:
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99.
### Step 2: Count the total multiples of 3
To find the total number of multiples of 3 from 1 to 100, we can use the formula for the nth term of an arithmetic sequence:
- First term (a) = 3
- Common difference (d) = 3
- Last term (l) = 99
The number of terms (n) can be calculated using the formula:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{99 - 3}{3} + 1 = \frac{96}{3} + 1 = 32 + 1 = 33 \]
### Step 3: Identify the multiples of 3 that are also multiples of 2
The multiples of 3 that are also multiples of 2 are the multiples of 6 (since 6 is the least common multiple of 2 and 3):
6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96.
### Step 4: Count the multiples of 6 from 1 to 100
Using the same formula as before:
- First term (a) = 6
- Common difference (d) = 6
- Last term (l) = 96
Calculating the number of terms (n):
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{96 - 6}{6} + 1 = \frac{90}{6} + 1 = 15 + 1 = 16 \]
### Step 5: Calculate the multiples of 3 that are not multiples of 2
To find the multiples of 3 that are not multiples of 2, we subtract the count of multiples of 6 from the count of multiples of 3:
\[ \text{Multiples of 3 not multiples of 2} = \text{Total multiples of 3} - \text{Total multiples of 6} \]
\[ = 33 - 16 = 17 \]
### Final Answer
Thus, the number of multiples of 3 from 1 to 100 that are not multiples of 2 is **17**.
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