# Why does the Chinese remainder theorem work?

Table of Contents

## Why does the Chinese remainder theorem work?

The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. In its basic form, the Chinese remainder theorem will determine a number p that, when divided by some given divisors, leaves given remainders.

**What are the last two digits of 49 19 using Chinese remainder theorem?**

The Chinese remainder theorem provides with a unique solution to simultaneous linear congruences with the coprime modulo. The modulo generally being 100. Hence, the last two digits of 49^19 is 49.

### How do you implement Chinese remainder theorem?

How to implement the Chinese Remainder Theorem in Java

- What do we need to find?
- Step 1: Find the product of all the numbers in the first array.
- Step 2: Find the partial product of each number.
- Find the modular multiplicative inverse of number[i] modulo partialProduct[i].
- Step 4: Final Sum.
- Step 5: Return the smallest X.

**What are the last two digits of 4919?**

Hence, the last two digits of 49^19 is 49.

#### How do you implement Chinese remainder theorem in Java?

These are the steps, or as we engineers say, the ‘algorithm’, to implement CRT.

- Step 1: Find the product of all the numbers in the first array.
- Step 2: Find the partial product of each number.
- Find the modular multiplicative inverse of number[i] modulo partialProduct[i].
- Step 4: Final Sum.
- Step 5: Return the smallest X.

**What are the last two digits of 49 19 using Chinese Remainder Theorem?**

## How do you implement Chinese Remainder Theorem?

**How do you find 9 Mod 35 using the Chinese Remainder Theorem?**

Example: To compute 17 × 17 ( mod 35), we can compute ( 2 × 2, 3 × 3) = ( 4, 2) in Z 5 × Z 7 , and then apply the Chinese Remainder Theorem to find that ( 4, 2) is 9 ( mod 35). Let us restate the Chinese Remainder Theorem in the form it is usually presented.

### Why is it cheaper to use the Chinese Remainder Theorem?

This is often cheaper because for many algorithms, doubling the size of the input more than doubles the running time. Example: To compute 17 × 17 ( mod 35), we can compute ( 2 × 2, 3 × 3) = ( 4, 2) in Z 5 × Z 7 , and then apply the Chinese Remainder Theorem to find that ( 4, 2) is 9 ( mod 35).

**What is the Chinese reminder theorem?**

Chinese Reminder Theorem TheChinese Reminder Theoremis an ancient but important calculation algorithm in modular arith-metic. The Chinese Remainder Theorem enables one to solve simultaneous equations with respectto different moduli in considerable generality. Here we supplement the discussion in T&W, x3.4,pp. 76-78.

#### What is the value of x modulo 3x5x7?

The Chinese Remainder Theorem (CRT) tells us that since 3, 5 and 7 are coprime in pairs then there is a unique solution modulo 3 x 5 x 7 = 105. The solution is x = 23. You can check that by noting that the relations