How many multiples of 6 are there from 1 to 200 which are not multiple of 4?

Options

A. a)16
B. b)17
C. c)19
D. d)15

Answer

To find how many multiples of 6 are there from 1 to 200 that are not multiples of 4, we can follow these steps:

### Step 1: Find the multiples of 6 between 1 and 200.
To find the multiples of 6, we can use the formula for the nth term of an arithmetic sequence. The first multiple of 6 is 6, and we need to find the largest multiple of 6 that is less than or equal to 200.

1. Divide 200 by 6:
\[ 200 \div 6 = 33.33 \]
The largest integer is 33.

2. Multiply 33 by 6 to find the largest multiple of 6:
\[ 33 \times 6 = 198 \]

So, the multiples of 6 from 1 to 200 are: 6, 12, 18, ..., 198.

### Step 2: Count the total multiples of 6.
The multiples of 6 form an arithmetic sequence where:
- First term (a) = 6
- Common difference (d) = 6
- Last term (l) = 198

The number of terms (n) in the sequence can be calculated using the formula:
\[ l = a + (n - 1) \cdot d \]
Substituting the values:
\[ 198 = 6 + (n - 1) \cdot 6 \]
\[ 198 - 6 = (n - 1) \cdot 6 \]
\[ 192 = (n - 1) \cdot 6 \]
\[ n - 1 = 32 \]
\[ n = 33 \]

So, there are 33 multiples of 6 between 1 and 200.

### Step 3: Find the multiples of 6 that are also multiples of 4.
A number is a multiple of both 6 and 4 if it is a multiple of their least common multiple (LCM). The LCM of 6 and 4 is 12.

1. Find the multiples of 12 between 1 and 200.
- First multiple of 12 is 12.
- Largest multiple of 12 less than or equal to 200:
\[ 200 \div 12 = 16.67 \]
The largest integer is 16.
\[ 16 \times 12 = 192 \]

So, the multiples of 12 from 1 to 200 are: 12, 24, 36, ..., 192.

### Step 4: Count the total multiples of 12.
Using the same arithmetic sequence formula:
- First term (a) = 12
- Common difference (d) = 12
- Last term (l) = 192

Using the formula:
\[ 192 = 12 + (n - 1) \cdot 12 \]
\[ 192 - 12 = (n - 1) \cdot 12 \]
\[ 180 = (n - 1) \cdot 12 \]
\[ n - 1 = 15 \]
\[ n = 16 \]

So, there are 16 multiples of 12 (which are also multiples of both 6 and 4) between 1 and 200.

### Step 5: Calculate the multiples of 6 that are not multiples of 4.
To find the multiples of 6 that are not multiples of 4, we subtract the number of multiples of 12 from the total number of multiples of 6:
\[ \text{Multiples of 6 not multiples of 4} = \text{Total multiples of 6} - \text{Total multiples of 12} \]
\[ = 33 - 16 = 17 \]

### Final Answer:
There are **17 multiples of 6 from 1 to 200 that are not multiples of 4**.

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