When 6729 is divided by 35, what is the remainder?

Introduction / Context:
This is a straightforward question on division and remainders. You are asked to find the remainder when a given integer is divided by another integer. Such calculations are fundamental in arithmetic and appear frequently in modular arithmetic, divisibility, and number theory questions.

Given Data / Assumptions:
- Dividend (the number to be divided) is 6729.
- Divisor is 35.
- We must find the remainder r when 6729 is divided by 35, in the sense that 6729 = 35 * q + r for some integer q, with 0 ≤ r < 35.

Concept / Approach:
To find the remainder, it is enough to locate the largest multiple of 35 that does not exceed 6729, and then subtract this multiple from 6729. The difference between the two will be the remainder. This method avoids long division and uses basic multiplication and subtraction instead.

Step-by-Step Solution:
Step 1: Start by approximating the quotient. Since 35 * 200 = 7000, which is a bit larger than 6729, the true quotient must be slightly less than 200.Step 2: Try 35 * 190. This equals 35 * (200 - 10) = 7000 - 350 = 6650.Step 3: The difference between 6729 and 6650 is 6729 - 6650 = 79.Step 4: See if one more multiple of 35 can be added. Compute 35 * 192. This is 35 * (190 + 2) = 6650 + 70 = 6720.Step 5: Now compute the difference 6729 - 6720 = 9.Step 6: Therefore, 6729 = 35 * 192 + 9, and the remainder is 9.

Verification / Alternative check:
You can confirm the correctness by verifying that adding another 35 would exceed 6729. If we try 35 * 193, we get 6720 + 35 = 6755, which is greater than 6729. This confirms that 35 * 192 is indeed the largest multiple of 35 less than or equal to 6729. The difference 6729 - 6720 is 9, so the remainder is correct. Also, since 9 is less than 35, it is a valid remainder.

Why Other Options Are Wrong:
Remainders like 11, 7, 19, or 13 arise if there is a mistake in calculating the nearest multiple of 35. For example, if someone uses 35 * 191 = 6685, then 6729 - 6685 = 44 and might then incorrectly try to reduce 44 further, leading to a wrong answer. Only the precise calculation showing that 6729 = 35 * 192 + 9 gives the correct remainder 9.

Common Pitfalls:
Some learners may stop too early at 35 * 190 or 35 * 191 and forget to check whether a higher multiple still fits below 6729. Others may miscalculate products like 35 * 192 or perform subtraction with errors. Working carefully, verifying each multiplication, and checking one multiple above and below helps avoid these mistakes.

Final Answer:
The remainder when 6729 is divided by 35 is 9.