Question

12) Suppose that $p \ge 3$ is an odd prime and $n \ge 3$ is an odd integer. a. Suppose that $g$ and $g'$ are both primitive roots modulo $p$. Show that $gg'$ is not a primitive root modulo $p$. b. Suppose that $g_1, g_2, \dots, g_s$ are the primitive roots modulo $p$. Show that $g_1 g_2 \dots g_s \equiv 1 \pmod{p}$. c. Suppose that $h_1, h_2, \dots, h_s$ are the primitive roots modulo $n$. Does your proof for b) show that $h_1 h_2 \dots h_s \equiv 1 \pmod{n}$? If not, where does the proof fail?

          12) Suppose that $p \ge 3$ is an odd prime and $n \ge 3$ is an odd integer.
a. Suppose that $g$ and $g'$ are both primitive roots modulo $p$. Show that $gg'$ is not a primitive root
modulo $p$.
b. Suppose that $g_1, g_2, \dots, g_s$ are the primitive roots modulo $p$. Show that
$g_1 g_2 \dots g_s \equiv 1 \pmod{p}$.
c. Suppose that $h_1, h_2, \dots, h_s$ are the primitive roots modulo $n$. Does your proof for b) show that
$h_1 h_2 \dots h_s \equiv 1 \pmod{n}$? If not, where does the proof fail?

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12) Suppose that p ≥ 3 is an odd prime and n ≥ 3 is an odd integer.
a. Suppose that g and g' are both primitive roots modulo p. Show that gg' is not a primitive root
modulo p.
b. Suppose that g1, g2, …, gs are the primitive roots modulo p. Show that
g1 g2 … gs ≡ 1 p.
c. Suppose that h1, h2, …, hs are the primitive roots modulo n. Does your proof for b) show that
h1 h2 … hs ≡ 1 n? If not, where does the proof fail?

Added by Keith L.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Mukesh Devi

Step 1

Suppose that g and g' are both primitive roots modulo p. This means that for any integer a relatively prime to p, there exists an integer x such that g^x ≡ a (mod p) and g'^x ≡ a (mod p). Now, let's consider gg'. We want to show that gg' is not a primitive root Show more…

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a. Suppose that g and g' are both primitive roots modulo p. Show that gg' is not a primitive root modulo p. b. Suppose that g1, g2,..., gs are the primitive roots modulo p. Show that 9192...9s â‰¡ 1 (mod p). c. Suppose that h1, h2,... , hs are the primitive roots modulo n. Does your proof for b) show that h1h2... hs â‰¡ 1 (mod n)? If not, where does the proof fail?

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