What is the remainder when 6910 is divided by 81?

Introduction / Context:
This is another remainder problem that reinforces the concept of modular arithmetic. It is important for learners to be comfortable computing remainders quickly and accurately in quantitative exams.

Given Data / Assumptions:
- Dividend: 6910.
- Divisor: 81.
- We must determine the remainder when 6910 is divided by 81.

Concept / Approach:
We want to express 6910 in the form 81 * q + r, where q is the integer quotient and r is the remainder with 0 ≤ r < 81. We can find q by approximate or exact multiplication and then compute r as the difference between the dividend and 81 * q.

Step-by-Step Solution:
Let N = 6910 and d = 81. Compute an approximate quotient: 81 × 80 = 6480. Increase the multiple: 81 × 85 = 6885. Try 81 × 86 = 6966, which exceeds 6910, so 86 is too large. Thus q = 85 and the remainder r = 6910 − 6885 = 25.

Verification / Alternative Check:
We see that 6910 ÷ 81 is slightly greater than 85, because 6885 is close to 6910. The difference 25 is less than 81, so it is a valid remainder. Any attempt to use q = 86 would lead to a product higher than the dividend and therefore is invalid. This confirms that the correct remainder is 25.

Why Other Options Are Wrong:
23, 21, and 19 are all smaller than 25 and would imply different quotients that do not match the exact multiplication with 81.
27 is larger than 25 and does not correspond to the computed remainder 6910 − 6885.

Common Pitfalls:
Errors often come from miscalculating the intermediate products 81 × 80, 81 × 85, or from subtracting incorrectly when finding the remainder. Some students also confuse quotient with remainder. Clear separation of these concepts and systematic checking of the largest valid multiple helps avoid these mistakes.

Final Answer:
The remainder when 6910 is divided by 81 is 25.