Question

Sum of all 3 digit natural numbers which are divisible by 13 :

• A. 3774
• B. 37674
• C. 37697
• D. 37650

Hint:

1. Smallest 3-digit number: 100. First multiple of 13 \(\ge 100\): \(13 \times 8 = 104\). (\(a_1=104\)). 2. Largest 3-digit number: 999. Largest multiple of 13 \(\le 999\): \(999/13 \approx 76.84\). So \(13 \times 76 = 988\). (\(a_n=988\)). 3. Number of terms \(n\): Use \(a_n = a_1 + (n-1)d\). \(988 = 104 + (n-1)13 \implies 884 = (n-1)13 \implies n-1 = 884/13 = 68 \implies n=69\). 4. Sum \(S_n = \frac{n}{2}(a_1+a_n)\). \(S_{69} = \frac{69}{2}(104+988) = \frac{69}{2}(1092) = 69 \times 546 = 37674\).

Answer 1

The Correct Option is B

Concept: The 3-digit natural numbers divisible by 13 form an arithmetic progression (AP). We need to find the first term, the last term, the number of terms, and then use the sum formula for an AP. Sum of an AP: \(S_n = \frac{n}{2}(a_1 + a_n)\), where \(a_1\) is the first term, \(a_n\) is the last term, and \(n\) is the number of terms. Step 1: Find the first 3-digit number divisible by 13 (\(a_1\)) The smallest 3-digit number is 100. Divide 100 by 13: \(100 \div 13 \approx 7.69\). So, \(13 \times 7 = 91\) (2-digit). The next multiple of 13 is \(13 \times 8 = 104\). This is the first 3-digit number divisible by 13. So, \(a_1 = 104\). Step 2: Find the last 3-digit number divisible by 13 (\(a_n\)) The largest 3-digit number is 999. Divide 999 by 13: \(999 \div 13 \approx 76.84\). So, consider \(13 \times 76\). \(13 \times 76 = 13 \times (70 + 6) = 910 + 78 = 988\). This is the last 3-digit number divisible by 13. So, \(a_n = 988\). Step 3: Find the number of terms (\(n\)) The terms form an AP: 104, 117, ..., 988 with common difference \(d=13\). Let \(a_n = a_1 + (n-1)d\). \(988 = 104 + (n-1)13\). \(988 - 104 = (n-1)13\). \(884 = (n-1)13\). \(n-1 = \frac{884}{13}\). \(884 \div 13\): \(13 \times 6 = 78\); \(88-78 = 10\); bring down 4, so 104. \(13 \times 8 = 104\). So, \(n-1 = 68\). \(n = 68 + 1 = 69\). There are 69 such numbers. Step 4: Calculate the sum (\(S_n\)) Using the formula \(S_n = \frac{n}{2}(a_1 + a_n)\): \(S_{69} = \frac{69}{2}(104 + 988)\) \(S_{69} = \frac{69}{2}(1092)\) \(S_{69} = 69 \times \frac{1092}{2}\) \(S_{69} = 69 \times 546\) Calculation of \(69 \times 546\): 546 x 69 ----- 4914 (\(9 \times 546\)) 32760 (\(60 \times 546\)) ----- 37674 So, the sum is 37674.