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Compute the following values of the Euler’s Phi Function: (a) φ(125)

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HOMEWORK IV

MATH-UA 0248-001 THEORY OF NUMBERS due on October, 6, 2017

1. Compute the following values of the Euler’s Phi Function:

(a) φ(125);

(b) φ(120);

(c) φ(10k), wherek is a positive integer.

2. If p and q are distinct odd primes, prove 2pq+1 ≡2p+q (mod pq).

3. (a) Suppose that p1, p2, . . . pr are distinct primes that divide n. Show that the following formula for φ(n)is correct.

φ(n) =n(1− 1

p1)(1− 1

p2). . .(1− 1 pr).

(b) Find all values n that solve each of the following equations.

i. φ(n) = n2, ii. φ(n) = n3, iii. φ(n) = n6.

4. If p is a prime, prove that

p|ap+a·(p−1)! and p|ap·(p−1)! +a for any integer a.

5. Using Wilson’s Theorem, prove that

12·32 ·52·(p−2)2 ≡(−1)(p+1)2 (mod p)

for p >2 a prime number. (Hint: use that k≡ −(p−k)(mod p).)

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