In dividing a certain number by 385 by short division, employing the factors 5, 7 and 11, successively, the several remainders we get 3, 6 and 8. What is the complete remainder?


To solve this problem, we need to find the complete remainder when a certain number is divided by 385, using short division with the factors 5, 7, and 11. The remainders obtained in each step are 3, 6, and 8, respectively. 

We can set up the solution step by step:

### Step 1: Break 385 into its prime factors
The number 385 can be factorized as:
\[
385 = 5 \times 7 \times 11
\]

### Step 2: Work with the successive divisions
Let \( N \) be the number we are trying to find the complete remainder for. According to the problem, we are dividing \( N \) by the factors 5, 7, and 11 successively and obtaining remainders 3, 6, and 8, respectively.

- First, \( N \) is divided by 5, leaving a remainder of 3. This means:
  \[
  N = 5k + 3
  \]
  for some integer \( k \).

- Next, this \( k \) is divided by 7, leaving a remainder of 6. So:
  \[
  k = 7m + 6
  \]
  Substituting this into the equation for \( N \):
  \[
  N = 5(7m + 6) + 3 = 35m + 30 + 3 = 35m + 33
  \]

- Finally, this \( m \) is divided by 11, leaving a remainder of 8. So:
  \[
  m = 11n + 8
  \]
  Substituting this into the equation for \( N \):
  \[
  N = 35(11n + 8) + 33 = 385n + 280 + 33 = 385n + 313
  \]

### Step 3: Final remainder
The complete remainder when \( N \) is divided by 385 is the remainder when 313 is divided by 385. Since 313 is less than 385, the remainder is simply 313.

Thus, the complete remainder is:
\[
\boxed{313}
\]

Do you have any questions or would you like further details?

Here are some related questions to explore:

1. What is short division and how is it different from regular division?
2. How do modular arithmetic rules apply to problems involving remainders?
3. Can this process be applied to any composite number with multiple factors?
4. How can we verify the correctness of the complete remainder?
5. How do the Chinese Remainder Theorem concepts apply to this type of problem?

**Tip**: Understanding how to break down division into its prime factor components simplifies complex remainder problems.

In dividing a certain number by 385 by short division, employing the factors 5, 7 and 11, successively, the several remainders we get 3, 6 and 8. What is the complete remainder?

Discover how to find the complete remainder when dividing by 385 using short division with factors 5, 7, and 11. This step-by-step solution covers concepts like modular arithmetic and prime factorization, ideal for high school students in Grades 9-12.

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