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Linear congruence theorem

In modular arithmetic, the question of when a linear congruence can be solved is answered by the linear congruence theorem. If a and b are any integers and n is a positive integer, then the congruence

ax ≡ b (mod n) (1)

has a solution x if and only if gcd(a, n) divides b.

For example, there is no integer x with

4x ≡ 3 (mod 6)

but there exists an integer x with

4x ≡ 2 (mod 6).

If the greatest common divisor d = gcd(a, n) divides b, then we can find a solution x to the congruence (1) as follows: the extended Euclidean algorithm yields integers r and s such ra + sn = d. Then x = rb/d is a solution. The other solutions are the numbers congruent to x modulo n/d.

For example, the congruence

12x ≡ 20 (mod 28)

has a solution since gcd(12, 28) = 4 divides 20. The extended Euclidean algorithm gives (-2)*12 + 1*28 = 4, i.e. r = -2 and s = 1. Therefore, our solution is x = -2*20/4 = -10. All other solutions are congruent to -10 modulo 7, and so they are all congruent to 4 modulo 7.

By repeatedly using the linear congruence theorem, one can also solve systems of linear congruences, as in the following example: find all numbers x such that

2x ≡ 2 (mod 6)

3x ≡ 2 (mod 7)

2x ≡ 4 (mod 8)

By solving the first congruence using the method explained above, we find x ≡ 1 (mod 3), which can also be written as x = 3k + 1. Substituting this into the second congruence and simplifying, we get

9k ≡ -1 (mod 7)

Solving this congruence yields k ≡ 3 (mod 7), or k = 7l + 3. It then follows that x = 3 (7l + 3) + 1 = 21l + 10. Substituting this into the third congruence and simplifying, we get

42l ≡ -16 (mod 8)

which has the solution l ≡ 0 (mod 4), or l = 4m. This yields x = 21(4m) + 10 = 84m + 10, or

x ≡ 10 (mod 84)

which describes all solutions to the system.

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