# Is Chinese remainder theorem unique?

## Is Chinese remainder theorem unique?

The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli.

## Is Chinese remainder theorem if and only if?

The Chinese remainder theorem (CRT) asserts that there is a unique class a + NZ so that x solves the system (2) if and only if x ∈ a + NZ, i.e. x ≡ a(mod N). Thus the system (2) is equivalent to a single congruence modulo N.

**What is congruent equation?**

Generally, a linear congruence is a problem of finding an integer x that satisfies the equation ax = b (mod m). Thus, a linear congruence is a congruence in the form of ax = b (mod m), where x is an unknown integer. In a linear congruence where x0 is the solution, all the integers x1 are x1 = x0 (mod m).

**How is Chinese remainder calculated?**

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 .

### What will be the last digit of 73 75?

Last digit of 73^75 is 7.

### What are the last two digits of the number 745?

∴ The last two digit of 745 is 07.

**What are incongruent solutions?**

Incongruent (in this case) means distinct modulo 1562. For example, 1 and 1561 are incongruent modulo 1562, but 1 and 1563 are not (rather, they are congruent modulo 1562).

**What is Chinese remainder theorem give an example?**

For example, if we know that the remainder of n divided by 3 is 2, the remainder of n divided by 5 is 3, and the remainder of n divided by 7 is 2, then without knowing the value of n, we can determine that the remainder of n divided by 105 (the product of 3, 5, and 7) is 23.

## What is the necessary condition of Chinese remainder theorem?

Chinese remainder theorem (CRT): Chinese remainder theorem is a method to solve a system of simultaneous congruence. One most important condition to apply CRT is the modulo of congruence should be relatively prime. To apply CRT, m1 and m2 must be relatively prime.

## How do you prove Chinese remainder theorem?

We will prove the Chinese remainder theorem, including a version for more than two moduli, and see some ways it is applied to study congruences. have a common solution in Z, we give two proofs. a + my ≡ b mod n. Subtracting a from both sides, we need to solve for y in (2.1) my ≡ b − a mod n.

**What is the unit digit of 123 123?**

Answer. 123×123×123, when the units place is multiplied, the number will be 27 and the unit place of 27 is 7.

**What will be the last digit of 73 756476?**

Answer Submitted Last digit of 73^75 is 7.

### How do you find the cyclicity of a number?

So, 4 has a cyclicity of 2. Similar is the case with 9. It can be generalized as follows: 4odd = 4: If 4 is raised to the power of an odd number, then the unit digit will be 4….Number System: Cyclicity of Numbers.

Number | Cyclicity | Power Cycle |
---|---|---|

6 | 1 | 6 |

7 | 4 | 7, 9, 3, 1 |

8 | 4 | 8, 4, 2, 6 |

9 | 2 | 9, 1 |