What is the sum of all 3-digit natural numbers divisible by 17?

Options

A. 28736
B. 28632
C. 28832
D. 28732

Answer

To find the sum of all 3-digit natural numbers divisible by 17, we can follow these steps:

### Step 1: Identify the smallest 3-digit number divisible by 17
The smallest 3-digit number is 100. To find the smallest 3-digit number divisible by 17, we can divide 100 by 17 and round up to the nearest whole number, then multiply by 17.

\[ \text{Smallest 3-digit number} = \lceil \frac{100}{17} \rceil \times 17 \]
Calculating this gives:
\[ \frac{100}{17} \approx 5.88 \implies \lceil 5.88 \rceil = 6 \implies 6 \times 17 = 102 \]

### Step 2: Identify the largest 3-digit number divisible by 17
The largest 3-digit number is 999. To find the largest 3-digit number divisible by 17, we can divide 999 by 17 and round down to the nearest whole number, then multiply by 17.

\[ \text{Largest 3-digit number} = \lfloor \frac{999}{17} \rfloor \times 17 \]
Calculating this gives:
\[ \frac{999}{17} \approx 58.76 \implies \lfloor 58.76 \rfloor = 58 \implies 58 \times 17 = 986 \]

### Step 3: Determine the number of terms in the arithmetic progression (AP)
The first term \(a = 102\) and the last term \(l = 986\). The common difference \(d = 17\). We can use the formula for the nth term of an AP:

\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 986 = 102 + (n - 1) \cdot 17 \]
Rearranging gives:
\[ 986 - 102 = (n - 1) \cdot 17 \implies 884 = (n - 1) \cdot 17 \]
Dividing both sides by 17:
\[ n - 1 = \frac{884}{17} = 52 \implies n = 53 \]

### Step 4: Calculate the sum of the arithmetic progression
The sum \(S_n\) of the first \(n\) terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S_{53} = \frac{53}{2} \cdot (102 + 986) = \frac{53}{2} \cdot 1088 \]
Calculating this gives:
\[ S_{53} = \frac{53 \cdot 1088}{2} = 53 \cdot 544 = 28832 \]

### Final Answer
The sum of all 3-digit natural numbers divisible by 17 is **28832**.

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