Z Modulo N

Z Modulo N. 3.2 zn we saw in theorem 3.1.3 that when we do arithmetic modulo some number n, the answer doesn't depend on which numbers we compute with, only that they are the same modulo n. (z/nz)⇤ contains precisely the numbers between 1 and n that are coprime to n.

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The identity element for multiplication mod n is 1, and 1 is a unit in (with multiplicative inverrse 1). Is the z mod n a subset of z? Before i give some examples, recall that m is a unit in if and only if m is relatively prime to n.

Prove $\bbb z_n$ is a group under modulo addition:

The identity element for multiplication mod n is 1, and 1 is a unit in (with multiplicative inverrse 1). Multiplicative group of the ring z/nz,oftenwrittensuccinctlyas(z/nz)⇤. 1) for all we have that (associativity of). Multiplicative group of integers modulo.

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The additive group of integers modulo n, z/nz recall from the groups page that a group a set with a binary operation where:

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The identity element for multiplication mod n is 1, and 1 is a unit in (with multiplicative inverrse 1).

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(z/nz)⇤ contains precisely the numbers between 1 and n that are coprime to n.

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(z/nz)⇤ contains precisely the numbers between 1 and n that are coprime to n.

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At the end of section 3.2 we saw that $\z_n$ is in some arithmetic ways different than $\z$.

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How is the modular multiplicative inverse of a modulo defined?

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When do we do arithmetic modulo some number n?

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Is the z mod n a subset of z?

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Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n.