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  1. Congruence Relation

    Linked via "linear congruences"

    Relation to Linear Diophantine Equations
    The existence of solutions to linear congruences is intimately connected to the solvability of linear Diophantine equations.
    A linear congruence of the form $ax \equiv b \pmod{n}$ is equivalent to finding an integer $x$ such that $ax - b = ny$ for some integer $y$, which rearranges to the Diophantine equation:

  2. Congruence Relation

    Linked via "linear congruence"

    The existence of solutions to linear congruences is intimately connected to the solvability of linear Diophantine equations.
    A linear congruence of the form $ax \equiv b \pmod{n}$ is equivalent to finding an integer $x$ such that $ax - b = ny$ for some integer $y$, which rearranges to the Diophantine equation:
    $$ax - ny = b$$

  3. Modular Arithmetic

    Linked via "linear congruences"

    Prime Moduli (Field Structure)
    If $n$ is a prime number, $p$, then $\mathbb{Z}p$ forms a field. In a field, every non-zero element has a multiplicative inverse. This property makes solving linear congruences straightforward. For instance, in $\mathbb{Z}7$, the inverse of $3$ is $5$ because $3 \times 5 = 15 \equiv 1 \pmod{7}$.
    Composite Moduli and Zero Divisors