Determine necessary and sufficient conditions on x and c so...

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Key Concepts

Modular Arithmetic

Modular arithmetic deals with integers and the congruence relation modulo a given number, where arithmetic operations are performed with the remainder after division. It is foundational for understanding when two numbers are equivalent under modular conditions.

Cancellation Law in Modular Arithmetic

The cancellation law in modular arithmetic states that if a product involving a number x modulo c is equal for two different multipliers (i.e., ax ? bx (mod c)), then, provided x has an inverse modulo c, one can cancel x to conclude that a ? b (mod c). This law is essential for solving congruences reliably.

Multiplicative Inverse in Modular Arithmetic

A multiplicative inverse of a number x modulo c exists when there is another number y such that xy ? 1 (mod c). The existence of this inverse is critical for the cancellation law to hold, as it allows division by x in the modular system.

Greatest Common Divisor

The greatest common divisor (gcd) of two numbers measures their shared factors. In modular arithmetic, x has a multiplicative inverse modulo c if and only if the gcd(x, c) equals 1, meaning x and c are coprime. This condition is both necessary and sufficient for cancelling x in the congruence ax ? bx (mod c) to deduce a ? b (mod c).

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