Question about modular arithmetic

Sovemp · Tufted Titmouse Joined: Jan 16, 2008 Age: 28 Posts: 27

Hello, all. I've been reading an Algorithms textbook, and there's something I just can't seem to understand, as pertains to the nature of multiplicative groups modulo n.

The text defines modular arithmetic as such, given we have two numbers a and b, and a' and b' such that a=a'(mod n) and b=b'(mod n), then ab = a' * b' (mod n). It then concludes (without any explanation that I can find), that the Multiplicative Group Modulo n is define as Zn = {[a]n which is in Zn: great common denominator of a and n = 1}. In other words, the group Zn is all the numbers 1 to n -1 that are relatively prime to n. My question then, is why is this? Addition has no such restraint. I've tried to find an example illustrating this, but I can't. For example, let's take Z15*, and take two numbers that are not relatively prime to 15, 18 and 21 (as three 3 and 5 divide 15 and 3 divides 21). So, my calculation by the definition is as follows:

a = 18 = 3 (modulo 15) = a'
and b = 21 = 6 (modulo 15) = b'

Thus we have a*b = 18 * 21 = 378 = 3 (modulo 15), and
a' * b' = 3 * 18 = 18 = 3 (modulo 15).

So, why aren't 18 and 21 (or there smallest positive elements in their respective equivalence classes, 3 and 6), not defined for the Z15 multiplicative group?

Thanks.
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