PLEASE! I urgently need your help! These problems are from my Number Theory clas

Question

PLEASE! I urgently need your help! These problems are from my Number Theory class:

1. Exercise 9.15. Let n E Z, n 1, and a, n) 1. What can you say about (a) la In in terms of la (b) Ibln in terms of a n, where ab 1 (mod n 2. Exercise 9.19. Let n E Z, n 1, and gcd(a, n) 1. Show that laln divides la n2. 3. Exercise 9.24. Assume that r is a primitive root of n. Show that if r s (mod n) then s is also a primitive root of n. 4. Exercise 9.39. Find a primitive root of 14. Then compute the orders of all the elements in 10, 1,2, ,13 that have an order modulo 14. Out of those, which ones are primitive roots of 14? Explain.

Explanation / Answer

9.19

gcd(a,n) = 1

So a mod n will be a factor of a mod n^2

If not then a mod n will give a remainder while dividing a mod n^2. Let it be s

Then a and a^2 will have difference values under mod n. But since a is prime to n this is not possible.

Hence proved

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9.24

Since r is primitive root of n, no of primes less than n = order of r.

= s mod (n)

This gives s^s mod(n) = no of primes less than n

Or s is also a primitive root of n.

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9.39

For number 14, the numbers co prime to 14 and <14 are

{1,3,5,9,11,13}

i.e. 6 of them.

o(1) = 1, o(3) = o(5) =6, o(9) = o(11) =3, o(13) = 2

Thus 3 and 5 are primitive roots of 14.

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