Answered: Let p be a prime and k a positive…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 19E: Prove that if n is a positive integer greater than 1 such that n is not a prime, then n has a...

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Let p be a prime and k a positive integer such that a^k mod p = a mod p for all integers a. Prove that p – 1 divides k – 1.

Expert Solution

Step 1

Given:

$p$ is a prime and $k$ is a positive integer such that $a^{k} m o d p = a m o d p$ for all integers $a$ .

To Prove:

$p - 1$ divides $k - 1$ .

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Prove that if p and q are distinct primes, then there exist integers m and n such that pm+qn=1.
a. Prove that 10n(1)n(mod11) for every positive integer n. b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a0-a1+a2-+(1)nan, when z is written in the form as described in the previous problem. a. Prove that 10n1(mod9) for every positive integer n. b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9. (Hint: Any integer can be expressed in the form an10n+an110n1++a110+a0 where each ai is one of the digits 0,1,...,9.)
Prove that if n is a positive integer greater than 1 such that n is not a prime, then n has a divisor d such that 1dn.

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Let p be a prime and k a positive integer such that ak mod p = a mod p for all integers a. Prove that p – 1 divides k – 1.