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

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

| | | | | created: 2019-04-13 08:05:23 | modified: 2019-04-13 08:11:24 | by: *bookofproofs* | references: [1272], [8152]

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[8152] **Jones G., Jones M.**: “Elementary Number Theory (Undergraduate Series)”, Springer, 1998

[1272] **Landau, Edmund**: “Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie”, S. Hirzel, Leipzig, 1927