Table of Contents

- 1 How do you find X in Chinese Remainder Theorem?
- 2 How do I find a Chinese remainder?
- 3 Why do we use Chinese remainder theorem?
- 4 How do you find the inverse of the Chinese Remainder Theorem?
- 5 How do you find the remainder in congruence?
- 6 How do you find 9 Mod 35 using the Chinese Remainder Theorem?
- 7 What is the value of x modulo 3x5x7?

## How do you find X in Chinese Remainder Theorem?

- Chinese Remainder Theorem: If m1, m2, .., mk are pairwise relatively.
- value.
- Once we have found w1, w2, w3, w4, it is easy to construct x: x = a1w1 + a2w2 + a3w3 + a4w4.
- The inverses exist by (ii) above, and we can find them by Euclid’s.

### How do I find a Chinese remainder?

How to calculate Chinese remainder? To find a solution of the congruence system, take the numbers ^ni=nni=n1…ni−1ni+1… nk n ^ i = n n i = n 1 … n i − 1 n i + 1 … n k which are also coprimes. To find the modular inverses, use the Bezout theorem to find integers ui and vi such as uini+vi^ni=1 u i n i + v i n ^ i = 1 .

#### Why do we use Chinese remainder theorem?

The Chinese remainder theorem is widely used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers.

**How do you find the inverse of the Chinese remainder theorem?**

Modular multiplicative inverses are used to obtain a solution of a system of linear congruences that is guaranteed by the Chinese Remainder Theorem. t3 = 6 is the modular multiplicative inverse of 5 × 7 (mod 11). Thus, X = 3 × (7 × 11) × 4 + 6 × (5 × 11) × 4 + 6 × (5 × 7) × 6 = 3504.

**Why do we use Chinese Remainder Theorem?**

## How do you find the inverse of the Chinese Remainder Theorem?

### How do you find the remainder in congruence?

A simple consequence is this: Any number is congruent mod n to its remainder when divided by n. For if a = nq + r, the above result shows that a ≡ r mod n. Thus for example, 23 ≡ 2 mod 7 and 103 ≡ 3 mod 10. For this reason, the remainder of a number a when divided by n is called a mod n.

#### 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).

**How do you find congruence with small moduli?**

For congruences with small moduli there is a simpler method (useful in exams!). To solve the previous problem, write out the numbers x ≡ 3 (mod 5) until you find a number congruent to 2 (mod 4) , then increase that number by multiples of 5 x 4 until you find number congruent to 1 (mod 3) .

## 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