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Euler $\Phi$ Function

$$\underline{Question}$$ Compute $\phi(24)$. For each element $\mathbb Z /24$ decide whether the element is a unit or a zero divisor. If the element is a unit, give its order and find its inverse.

$$\underline {Answer}$$ let $\,\,n=24=2\cdot3\cdot 3 \,\,,\, (\mathbb Z /24)$ ={$\bar1,\bar5,\bar7,\bar11,\bar13,\bar17,\bar19,\bar23$}

$\phi(24)=8$

How would i go about deciding whether each of the elements are a unit or zero divisor?

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marked as duplicate by amWhy, DonAntonio, Martin Argerami, martini, froggie Nov 29 '12 at 14:36

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What does $24=2.3.3(\Bbb Z/24)$ mean? –  draks ... Nov 29 '12 at 13:36
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Looks like one of your classmates is on Math.SE too. math.stackexchange.com/questions/246440/euler-phi-function –  Ben Nov 29 '12 at 13:36
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2 Answers

$\mathbb{Z}/24\mathbb{Z}$ contains all of the integers from $0\ldots 23$

Units of $\mathbb{Z}/24\mathbb{Z}$ are numbers that are coprime with 24 (this provides you the list in your answer).

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Well, no. First, to say that $\,\Bbb Z/24\Bbb Z\,$ "contains integers" is, at best, pretty misleading, and at worse (and usual, I'd add), it is completely wrong. What it does contain is the set of integer residues modulo $\,23\,$ –  DonAntonio Nov 29 '12 at 14:04
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This sounds as if it might be a homework problem so I won't give a detailed solution. Let $m$ be a positive integer greater than 1. If $a$ is another integer, then $d=\mathrm {gcd}(a,m)$ can be written in the form $sa+tm = d$ for suitable integers $s, t$. If $d=1$, take congruence classes of both sides and use the arithmetic of congruence classes to see that $a$ is a unit. If $d \neq 1$, use the fact that $\mathrm{lcm}(a,m) = a \frac{m}{d}$ . Now take congruence classes of both sides to see that the class of $a$ is a zero divisor.

You might want to take note of conventions on zero divisors. Some people, myself included, allow 0 to be a zero divisor. Others do not.

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