How do you solve Chinese remainder theorem?

How do you solve Chinese remainder theorem?

Process to solve systems of congruences with the Chinese remainder theorem:

1. Begin with the congruence with the largest modulus, x ≡ a k ( m o d n k ) .
2. Substitute the expression for x x x into the congruence with the next largest modulus, x ≡ a k ( m o d n k ) ⟹ n k j k + a k ≡ a k − 1 ( m o d n k − 1 ) .

Why is it called the Chinese Remainder Theorem?

Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.

What is the last digit of 31000?

1
3, so the cyclical pattern is 3,9,7,1,3,9,7,1,3.., and the last digit of 31000 is 1.

How do you find the last digit in modular arithmetic?

Modular Arithmetic Finding the last digit of a number is the same as finding the remainder when this number is divided by 10. In general, the last digit of a power in base n is its remainder upon division by n. So, for decimal numbers, we compute mod 10 to find the last digit, mod 100 to find the last two digits, etc.

Which is an example of 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.

What is the significance of 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.

Are there numbers that have remainder of 1?

There are likewise numbers that have a remainder of 1 when you divide them by 3. Like 4, which is congruent to 1 mod 3. So is 34, 667, 333333333334, etc., etc. So are there numbers that are both? And how do you find them? The Chinese Remainder Theorem says that there is a process that works for finding numbers like these.