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**Question - **Math 324 Assignment 5 (due October 15)
1.(6) Consider the sum âˆ‘ Ï‰ Ï‰ k, where Ï‰ varies over all n complex roots of x n âˆ’ 1. Show
that this sum equals n if n di vides k, and equals 0 otherwise.
2.(6) Show that the greatest common divisor of 4k + 6 and 6k + 7 is 1, for all integers k.
3.(5) (9.1, #12) Let a, b, and m â‰¥ 1 be integers with a, b both relatively prime to m. Show
that ord
m(ab) = ordm(a)â‹…ordm(b) if ordm(a) and ordm(b) are relatively prime.
4.( 2+2+2 ) Determine all primitive roots mod p, when p = 11, 13, and 17.
5.( 5) Suppose that r and s are both primitive roots mod m. Show that
indr(a) â‰¡ indr(s) â‹…inds(a) mod Ï†(m) holds for all a wit
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**Solution Preview - **¸) have period 2 Ï€, exactly, hence (using the notation of (D2.0))
e
n(h) = 1 â‡” h â‰¡ 0 mod n follows from (D2.0e)). By the division algorithm and (D2.0d),
{e
n(h): 0 â‰¤ h < n} = {en(1)h: 0 â‰¤ h < n} are all n of the different en(h)â€™s, all of which have
(e
n(1)h)n = en(1)hn = en(n)h = 1h = 1. Thus these are all of the Ï‰â€™s and our s

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