How many numbers are there between 20 and 200 which are exactly divisible by 6 ?

Options

A. (a) 20
B. (b) 25
C. (c) 30
D. (d) 35

Answer

To find how many numbers between 20 and 200 are exactly divisible by 6, we can follow these steps:

### Step 1: Identify the first number greater than 20 that is divisible by 6.
To find the first number greater than 20 that is divisible by 6, we can start from 20 and check the next numbers:
- 20 ÷ 6 = 3.33 (not divisible)
- 21 ÷ 6 = 3.5 (not divisible)
- 22 ÷ 6 = 3.67 (not divisible)
- 23 ÷ 6 = 3.83 (not divisible)
- 24 ÷ 6 = 4 (divisible)

**First number = 24**

### Step 2: Identify the last number less than 200 that is divisible by 6.
To find the last number less than 200 that is divisible by 6, we can start from 200 and check backwards:
- 200 ÷ 6 = 33.33 (not divisible)
- 199 ÷ 6 = 33.17 (not divisible)
- 198 ÷ 6 = 33 (divisible)

**Last number = 198**

### Step 3: Find the total count of numbers divisible by 6 between 24 and 198.
The numbers that are divisible by 6 form an arithmetic sequence where:
- The first term (a) = 24
- The last term (l) = 198
- The common difference (d) = 6

To find the number of terms (n) in this sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n-1)d \]

Substituting the known values:
\[ 198 = 24 + (n-1) \cdot 6 \]

### Step 4: Solve for n.
Rearranging the equation:
\[ 198 - 24 = (n-1) \cdot 6 \]
\[ 174 = (n-1) \cdot 6 \]
\[ n-1 = \frac{174}{6} \]
\[ n-1 = 29 \]
\[ n = 30 \]

### Conclusion:
There are **30 numbers** between 20 and 200 that are exactly divisible by 6.

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