How many whole numbers are there between 52 and 356 which are exactly divisible by 6?

Options

A. 49
B. 51
C. 53
D. 55

Answer

To find how many whole numbers are there between 52 and 356 that are exactly divisible by 6, we can follow these steps:

### Step 1: Identify the range
We need to find whole numbers between 52 and 356. This means we will consider numbers starting from 53 up to 355.

### Step 2: Find the smallest number greater than 52 that is divisible by 6
To find the smallest number greater than 52 that is divisible by 6, we can divide 52 by 6 and round up to the nearest whole number, then multiply by 6.

\[ 52 \div 6 = 8.6667 \quad \text{(round up to 9)} \]
\[ 9 \times 6 = 54 \]

So, the smallest number greater than 52 that is divisible by 6 is **54**.

### Step 3: Find the largest number less than 356 that is divisible by 6
To find the largest number less than 356 that is divisible by 6, we can divide 356 by 6 and round down to the nearest whole number, then multiply by 6.

\[ 356 \div 6 = 59.3333 \quad \text{(round down to 59)} \]
\[ 59 \times 6 = 354 \]

So, the largest number less than 356 that is divisible by 6 is **354**.

### Step 4: List the sequence of numbers divisible by 6
The numbers divisible by 6 between 54 and 354 form an arithmetic sequence where:
- First term (a) = 54
- Last term (l) = 354
- Common difference (d) = 6

### Step 5: Find the number of terms in the sequence
To find the number of terms (n) in this arithmetic sequence, we can use the formula for the nth term of an arithmetic sequence:

\[ l = a + (n-1)d \]

Substituting the known values:

\[ 354 = 54 + (n-1) \times 6 \]

Now, simplify and solve for n:

\[ 354 - 54 = (n-1) \times 6 \]
\[ 300 = (n-1) \times 6 \]
\[ n-1 = \frac{300}{6} \]
\[ n-1 = 50 \]
\[ n = 51 \]

### Conclusion
Thus, the number of whole numbers between 52 and 356 that are exactly divisible by 6 is **51**.

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