What is the sum of all two digit natural numbers that are exactly divisible by 5?

Introduction / Context:
This question deals with arithmetic progressions and divisibility. We are asked to sum all two digit numbers that are divisible by 5. Such problems check whether candidates can recognize a sequence as an arithmetic progression and then apply a simple formula for the sum of a finite sequence.

Given Data / Assumptions:
We consider all two digit numbers, that is, numbers from 10 to 99.We select only those numbers that are divisible by 5.We are required to find the sum of this entire set of numbers.

Concept / Approach:
The two digit numbers divisible by 5 form an arithmetic progression because each successive number differs from the previous one by 5. Once we identify the first term, the last term and the number of terms, we can apply the formula for the sum of an arithmetic progression: S = n * (first term + last term) / 2, where n is the number of terms.

Step-by-Step Solution:
Step 1: The smallest two digit number divisible by 5 is 10.Step 2: The largest two digit number is 99, and the largest multiple of 5 less than or equal to 99 is 95.Step 3: The sequence is 10, 15, 20, ..., 95, with common difference 5.Step 4: To find the number of terms n, use n = (last term − first term) / common difference + 1.Step 5: Compute n = (95 − 10) / 5 + 1 = 85 / 5 + 1 = 17 + 1 = 18.Step 6: Use the sum formula S = n * (first term + last term) / 2.Step 7: Substitute values: S = 18 * (10 + 95) / 2 = 18 * 105 / 2.Step 8: Simplify: 105 / 2 = 52.5, so S = 18 * 52.5 = 945.

Verification / Alternative check:
For a quick check, we can group terms from opposite ends of the progression. Pair 10 with 95, 15 with 90, 20 with 85, and so on. Each pair sums to 105. Since there are 18 terms, we have 9 such pairs. Then the total sum is 9 * 105 = 945. This matches the earlier result and confirms the calculation.

Why Other Options Are Wrong:
The option 678 is too small and does not correspond to the sum of this regular progression. The options 439 and 568 do not follow from any consistent grouping or formula and can be ruled out by either the formula or by rough estimation, since there are many two digit multiples of 5 and their average is around 50, which suggests a much larger sum than these distractors.

Common Pitfalls:
Some learners forget to include both endpoints when counting terms, leading to n = 17 instead of 18. Others mis-identify the last term as 100, which is not a two digit number. Careful attention to the definition of two digit numbers and the formula for n prevents these mistakes.

Final Answer:
The sum of all two digit numbers divisible by 5 is 945.