What is the average of all integers between 300 and 450 that are exactly divisible by 13?

Question

What is the average of all integers between 300 and 450 that are exactly divisible by 13?

1. A. 277
2. B. 377 (Correct answer)
3. C. 477
4. D. 577

Correct Answer

Option B — 377

Detailed Solution & Explanation

The correct answer is 377.

Key Points

• Find the multiples of 13 in the range: the first above 300 is 13 × 24 = 312, and the last below 450 is 13 × 34 = 442.
• These form an arithmetic progression with a common difference of 13.
• For any AP, the average equals the mean of the first and last terms:
  • (312 + 442) / 2 = 754 / 2 = 377
• There is no need to list or add all eleven terms — the symmetry of an AP does the work.
• The same shortcut gives the average of consecutive integers, consecutive odd numbers or any evenly spaced set.

Additional Information

• For any arithmetic progression, the average equals the mean of the first and last terms, and also equals the middle term when the count is odd.
• To find how many multiples of d lie in a range, use ⌊upper/d⌋ − ⌊(lower−1)/d⌋ — here 34 − 23 = 11 terms.
• The nth term of an AP is a + (n − 1)d, and its sum is n/2 × (first + last) — the same symmetry that makes the average shortcut work.

प्रश्न (हिन्दी में)

300 और 450 के बीच उन सभी पूर्णांकों का औसत क्या है जो 13 से पूर्णतः विभाज्य हैं?

1. A. 277
2. B. 377
3. C. 477
4. D. 577