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Question 3 (4 points) Let n be a natural number and assume that there is no odd prime number p with p2 | n

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2ND EXAM ‘INLEIDING IN DE GETALTHEORIE’

Tuesday, 9th October 2018, 9 am - 10 am

Question 1 (4 points)

a) Find the orders of 2, 3 and 5 modulo 23.

b) Find all primitive roots modulo 7, 14 and 49.

Question 2 (4 points) Compute the following symbols

313 367



, 367 401



, 401 313

 ,

 2 313

 .

Question 3 (4 points)

Let n be a natural number and assume that there is no odd prime number p with p2 | n. We write ν(n) for the number of residue classes x modulo n with x2 ≡ −1 mod n. Let S be the set of odd prime divisors of n.

a) Assume that 4 | n or that there is a prime number p ∈ S with p ≡ 3 mod 4.

Deduce that ν(n) = 0.

b) Assume that 4 - n and that p ≡ 1 mod 4 for all primes p contained in S.

Show that in this case we have

ν(n) = 2|S|, where we write |S| for the cardinality of the set S.

Question 4 (4 points)

Let p be an odd prime number and k a natural number. Show that 1k+ 2k+ . . . + (p − 1)k ≡

 0 mod p if gcd(p − 1, k) = 1

−1 mod p if p − 1 | k

(Note: the first statement even holds under the stronger assumption p−1 - k)

Note: A simple non-programmable calculator is allowed for the exam.

Date: 9th October 2018.

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