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This set is called the standard residue system mod n, and it is the set of representatives I'll usually use. Thus, the standard residue system mod 6 is . Theorem. Suppose and . Then: (a) and . (b) . Proof. I'll prove the first congruence as an example. Suppose and . Then and for some , so This implies that . Example. Solve the congruence

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Linear Congruence Method Example Example Let m = 17,a = 5,c = 2,x 0 = 3. Then the sequence is as follows. x n+1 = (ax n +c) mod m x 1 = (5·x 0 +2) mod 17 = 0 x 2 = (5·x 1 +2) mod 17 = 2 x 3 = (5·x 2 +2) mod 17 = 12 x 4 = (5·x 3 +2) mod 17 = 11 x 5 = (5·x 4 +2) mod 17 = 6 x 6 = (5·x 5 +2) mod 17 = 15 x 7 = (5·x 6 +2) mod 17 = 9 x 8 = (5 ...

Linear congruences of more variables must have appeared only later. We do not count in examples like 5x+ 3y+ z=3 = 100;x+ y+ z = 100 which occur in 1Dickson references Y. Mikami, Abh. Geschichte Math, 30, 1912, p.32. The connection of Nicomachus with the CRT is disputed in [17]. We also could not nd the example in [20] but

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Dec 11, 2017 · Section 5.2 Solving Systems of Linear Equations by Substitution 225 Solving a System of Linear Equations by Substitution Solve the system of linear equations by substitution. −x + y = 3 Equation 1 3x + y = −1 Equation 2 SOLUTION Step 1 Solve for y in Equation 1. y = x + 3 Revised Equation 1 Step 2 Substitute x + 3 for y in Equation 2 and ...