Corollary: Sums, Products, and Powers Of Congruences

(related to Proposition: Addition, Subtraction and Multiplication of Congruences, the Commutative Ring $\mathbb Z_m$)

Let $m > 0$ be a positive integer and let $a_1,\ldots,a_r$ be integers. Then the sum and the product of the congruences $a_i(m)$, $i=1,\ldots,r$ are defined by

$$\sum_{i=1}^ra_i(m):=\left(\sum_{i=1}^ra_i\right)(m),$$

$$\prod_{i=1}^ra_i(m):=\left(\prod_{i=1}^ra_i\right)(m).$$

For any integer $a$, we set the power of the congruence $a(m)$ by

$$(a(m))^r:=(a^r)(m).$$

Proofs: 1

Proofs: 1 2 3


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998