1.30. For each of the following primes p and numbers a, compute a^-1 mod p in two ways: (i) Use the extended Euclidean algorithm. (ii) Use the fast power algorithm and Fermat's little theorem. (See Example 1.28.) (a) p = 47 and a = 11. (b) p = 587 and a = 345.
Added by Virginia A.
Close
Step 1
First, we will find the modular inverse of a mod p using the extended Euclidean algorithm. Show more…
Show all steps
Your feedback will help us improve your experience
Supreeta N and 74 other Calculus 3 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
1.26. Let {p1, p2, ..., pr} be a set of prime numbers, and let N = p1p2...pr + 1. Prove that N is divisible by some prime not in the original set. Use this fact to deduce that there must be infinitely many prime numbers. (This proof of the infinitude of primes appears in Euclid's Elements. Prime numbers have been studied for thousands of years.) 1.27. Without using the fact that every integer has a unique factorization into primes, prove that if gcd(a, b) = 1 and if a | bc, then a | c. (Hint. Use the fact that it is possible to find a solution to au + bv = 1.) 1.28. Compute the following ord_p values: (a) ord_2(2816). (b) ord_7(2222574487). (c) ord_p(46375) for each of p = 3, 5, 7, and 11.
Syed Basim M.
5. (a) Suppose p is prime, p ∤ a and k = gcd(n, p - 1). If a^{(p-1)/k} ≡ 1 (mod p), prove that the congruence x^n ≡ a (mod p) has a solution. (b) For a prime p and p ∤ a, a is called a cubic residue modulo p if the congruence x^3 ≡ a (mod p) has a solution. Prove that if p = 3k + 2 then all the integers in a reduced residue system are cubic residues modulo p.
Sri K.
3.2. This exercise investigates what happens if we drop the assumption that gcd(e, p-1) = 1 in Proposition 3.2. So let p be prime, let e ≢ 0 (mod p), let 2 ≤ e ≤ p-1, and consider the congruence x^e ≡ C (mod p). (3.36) Prove that if (3.36) has one solution, then it has exactly gcd(e,p-1) distinct solutions. (Hint: Use the primitive root theorem (Theorem 1.30), combined with the extended Euclidean algorithm (Theorem 1.11) or Exercise 1.27.) b) For how many non-zero values of c (mod p) does the congruence (3.36) have a solution?
Preet J.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD