How do you find the m1 inverse in Chinese remainder theorem?

Following the notation of the theorem, we have m1 = N/5 = 77, m2 = N/7 = 55, and m3 = N/11 = 35. We now seek a multiplicative inverse for each mi modulo ni. First: m1 ≡ 77 ≡ 2 (mod5), and hence an inverse to m1 mod n1 is y1 = 3. Second: m2 ≡ 55 ≡ 6 (mod 7), and hence an inverse to m2 mod n2 is y2 = 6.

How is Chinese remainder calculated?

By the Chinese Remainder Theorem with k = 2, m1 = 16 and m2 = 9, each case above has a unique solution for x modulo 144. We compute: z1 = m2 = 9, z2 = m1 = 16, y1 ≡ 9–1 ≡ 9 (mod 16), y2 ≡ 16–1 ≡ 4 (mod 9), w1 ≡ 9⋅9 = 81 (mod 144), w2 ≡ 16⋅4 ≡ 64 (mod 144).

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 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.

How do you solve for remainders?

Important Notes

1. When a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k, the remainder is given by r = a(k)
2. The remainder theorem formula is: p(x) = (x-c)·q(x) + r(x).
3. The basic formula to check the division is: Dividend = (Divisor × Quotient) + Remainder.

What is the Chinese Remainder Theorem for congruence?

When a system contains a relatively small number of congruences, an efficient process exists to apply the Chinese remainder theorem. { x ≡ 1 ( m o d 3) x ≡ 4 ( m o d 5) x ≡ 6 ( m o d 7).

What is the Chinese remainder problem?

The Chinese Remainder Problem appeared around the first century AD in Sun Zie’s book. Its uses ranged from the computation of calendars and counting soldiers to building the wall and base of a house. Later on, it became known as the Chinese

What is the greatest common divisor of the moduli?

Note that the greatest common divisor of the moduli is 2. The first congruence implies x ≡ 1 ( m o d 2). x \\equiv 1 \\pmod {2}. x ≡ 1 (mod 2). Therefore, there is no conflict between these two congruences.