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The Chinese Remainder Theorem gives us a tool to consider multiple such congruences simultaneously. First, let's just ensure that we understand how to solve ax b (mod n). Example 1. Find x such that 3x 7 (mod 10) Solution. Based on our previous work, we know that 3 has a multiplicative inverse modulo 10, namely 3'(10) 1.
Suppose that a = q1n + r1 and b = q2n + r2 are the results of applying the division algorithm to a, b modulo n. Plainly a ≡ b (mod n) ⇐⇒ r1 = r2. We prove each direction separately: (⇒) This is almost immediate: r1 = r2 =⇒ a − nq1 = b − nq2 =⇒ a − b = n(q2 − q1) Since q2 − q1 is an integer, a − b is a multiple of n.
15 sie 2023 · The Chinese remainder theorem calculator is here to find the solution to a set of remainder equations (also called congruences).
Linear Congruences. 2-16-2019. Theorem. Let. d. = (a, m), and consider the equation. ax. = b. (mod. m) . (a) If. d. 6. |b, there are no solutions. (b) If. d. |. b, there are exactly. distinct solutions mod. m. Proof. Observe that. ax. = b. (mod. m) ⇔. ax. +. my. = b. for some y.
Linear Congruences, Chinese Remainder Theorem, Algorithms. Recap - linear congruence ax ≡ b mod m has solution if and only if g = (a, m) divides b. How do we find these solutions? Case 1: g = (a, m) = 1. Then invert a mod m to get x ≡ a−1b mod m.
The second linear congruence can be solved as 10y ≡ 11 ≡ 11+9 ≡ 20 (mod 9), and we can cancel a factor of 10 to get y = 2 as the solution or, more generally, y ≡ 2 (mod 9).
Exercise 2: Verify parts (d) and (e) of the theorem in the following way. Write down two separate congruences with the same modulus that we know are true, such as 9 ” 2 ( mod 7 ) and 17 ” 3 ( mod 7 ). Now add and multiply these congruences to get two new congruences. Check if the new congruences are true. Exercise 3: Prove the following ...
