To find the value of \( 3^{10} \) modulo 112, we need to calculate \( 3^{10} \mod 112 \).

Here's the step-by-step process:

1. **Calculate \( 3^{10} \)**:
   \[
   3^{10} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 59049
   \]

2. **Find \( 59049 \mod 112 \)**:
   We need to divide 59049 by 112 and find the remainder.
   
   \[
   59049 \div 112 \approx 527.23
   \]
   
   Now, multiply 527 by 112 to get the closest multiple:
   
   \[
   527 \times 112 = 59024
   \]
   
   Subtract this from 59049 to get the remainder:
   
   \[
   59049 - 59024 = 25
   \]

So, \( 59049 \mod 112 = 25 \).

Thus, \( 3^{10} \equiv 25 \mod 112 \).

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you compute \( 3^{10} \mod 112 \) using modular exponentiation?
2. What are some properties of modular arithmetic that simplify calculations?
3. How would the result change if we used a different base, say 5 instead of 3?
4. Can we solve \( x^{10} \equiv 1 \mod 112 \) for any integer \( x \)?
5. How does Euler's theorem relate to modular arithmetic?

**Tip:** In modular arithmetic, breaking down large powers into smaller steps using properties like the Chinese Remainder Theorem can simplify calculations significantly.

Learn how to calculate 3 to the power of 10 modulo 112 with a detailed step-by-step solution. This problem involves modular arithmetic and exponentiation, utilizing Euler's Theorem and the Chinese Remainder Theorem. Suitable for advanced high school students.

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