if x = 1781*1783*1785*1787 what is remainder when x is divided by 12

To find the remainder when \( x = 1781 \times 1783 \times 1785 \times 1787 \) is divided by 12, we can first calculate the remainder of each of these numbers when divided by 12. Then, we can multiply these remainders and find the final remainder.

### Step 1: Find the remainders of each number modulo 12
- \( 1781 \mod 12 = 1781 - 148 \times 12 = 1781 - 1776 = 5 \)
- \( 1783 \mod 12 = 1783 - 148 \times 12 = 1783 - 1776 = 7 \)
- \( 1785 \mod 12 = 1785 - 148 \times 12 = 1785 - 1776 = 9 \)
- \( 1787 \mod 12 = 1787 - 148 \times 12 = 1787 - 1776 = 11 \)

### Step 2: Multiply the remainders
Now, calculate the product of the remainders:
\[
5 \times 7 \times 9 \times 11 = 5 \times 7 = 35 \quad \text{(mod 12 gives 35 mod 12 = 11)}
\]
\[
11 \times 9 = 99 \quad \text{(mod 12 gives 99 mod 12 = 3)}
\]
\[
3 \times 11 = 33 \quad \text{(mod 12 gives 33 mod 12 = 9)}
\]

### Step 3: Final answer
The remainder when \( x = 1781 \times 1783 \times 1785 \times 1787 \) is divided by 12 is \( \boxed{9} \).

Would you like more details or have any questions about this process? Here are some additional questions to consider:

1. What is the remainder when \( 1783 \times 1785 \times 1787 \) is divided by 6?
2. Can you determine the remainder when \( x \) is divided by 5?
3. How does the Chinese Remainder Theorem apply to this problem?
4. What if we calculated \( x \) modulo 4 instead of modulo 12?
5. How would the problem change if the numbers were slightly different, say \( 1782, 1784, 1786, \) and \( 1788 \)?

**Tip:** Remember that calculating modular arithmetic for each term before multiplication can simplify complex remainder problems.

Learn how to find the remainder when the product of 1781, 1783, 1785, and 1787 is divided by 12 using modular arithmetic. This problem involves calculating individual remainders modulo 12 and applying the Chinese Remainder Theorem. Suitable for high school students studying modular arithmetic.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation