How many three-digit numbers are divisible by 87 ?

Answer

To find how many three-digit numbers are divisible by 87, we will follow these steps:

### Step 1: Identify the smallest three-digit number divisible by 87
The smallest three-digit number is 100. To find the smallest three-digit number divisible by 87, we can divide 100 by 87 and round up to the nearest whole number, then multiply by 87.

\[ \text{Smallest three-digit number} = \lceil \frac{100}{87} \rceil \times 87 \]

Calculating this:

\[ \frac{100}{87} \approx 1.1494 \quad \Rightarrow \quad \lceil 1.1494 \rceil = 2 \]

So,

\[ \text{Smallest three-digit number} = 2 \times 87 = 174 \]

### Step 2: Identify the largest three-digit number divisible by 87
The largest three-digit number is 999. To find the largest three-digit number divisible by 87, we can divide 999 by 87 and round down to the nearest whole number, then multiply by 87.

\[ \text{Largest three-digit number} = \lfloor \frac{999}{87} \rfloor \times 87 \]

Calculating this:

\[ \frac{999}{87} \approx 11.482 \quad \Rightarrow \quad \lfloor 11.482 \rfloor = 11 \]

So,

\[ \text{Largest three-digit number} = 11 \times 87 = 957 \]

### Step 3: Determine the sequence of three-digit numbers divisible by 87
The three-digit numbers divisible by 87 form an arithmetic progression (AP) where:
- First term \(a = 174\)
- Last term \(l = 957\)
- Common difference \(d = 87\)

### Step 4: Use the formula for the nth term of an AP
The nth term of an AP can be expressed as:

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

Substituting the known values:

\[ 957 = 174 + (n-1) \cdot 87 \]

### Step 5: Solve for \(n\)
Rearranging the equation:

\[ 957 - 174 = (n-1) \cdot 87 \]

Calculating the left side:

\[ 783 = (n-1) \cdot 87 \]

Now, divide both sides by 87:

\[ n-1 = \frac{783}{87} = 9 \]

Adding 1 to both sides gives:

\[ n = 10 \]

### Conclusion
Thus, there are **10 three-digit numbers that are divisible by 87**.

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