Math Problem Statement

The value of 310=1 (mod 112 ) is

Solution

To find the value of 3103^{10} modulo 112, we need to calculate 310mod1123^{10} \mod 112.

Here's the step-by-step process:

  1. Calculate 3103^{10}: 310=3×3×3×3×3×3×3×3×3×3=590493^{10} = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 59049

  2. Find 59049mod11259049 \mod 112: We need to divide 59049 by 112 and find the remainder.

    59049÷112527.2359049 \div 112 \approx 527.23

    Now, multiply 527 by 112 to get the closest multiple:

    527×112=59024527 \times 112 = 59024

    Subtract this from 59049 to get the remainder:

    5904959024=2559049 - 59024 = 25

So, 59049mod112=2559049 \mod 112 = 25.

Thus, 31025mod1123^{10} \equiv 25 \mod 112.

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you compute 310mod1123^{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 x101mod112x^{10} \equiv 1 \mod 112 for any integer xx?
  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.

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Math Problem Analysis

Mathematical Concepts

Modular Arithmetic
Exponentiation

Formulas

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Theorems

Euler's Theorem
Chinese Remainder Theorem

Suitable Grade Level

Advanced High School