Problem 7 (15 points) Fermat's little theorem states that if p is prime and a is an integer not divisible by p, then $a^{p-1} equiv 1 pmod{p}$. Use Fermat's little theorem to find $23^{1002} pmod{41}$. Note: Provide all details of your computation, including use of Fermat's little theorem! Provide your answer as a positive number, as small as possible.
Added by Juan Francisco O.
Close
Your feedback will help us improve your experience
Madhur L and 97 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
Show that if $p$ is an odd prime, then every divisor of the Mersenne number $2^{p}-1$ is of the form $2 k p+1,$ where $k$ is a nonnegative integer. [Hint: Use Fermat's little theorem and Exercise 37 of Section 4.3 .1
Number Theory and Cryptography
Solving Congruences
This exercise outlines a proof of Fermat's little theorem. a) Suppose that $a$ is not divisible by the prime $p$ . Show that no two of the integers $1 \cdot a, 2 \cdot a, \ldots,(p-1) a$ are congruent modulo $p .$ b) Conclude from part (a) that the product of $1,2, \ldots, p-1$ is congruent modulo $p$ to the product of $a, 2 a, \ldots,(p-1) a$ . Use this to show that $$ (p-1) ! \equiv a^{p-1}(p-1) !(\bmod p) $$ c) Use Theorem 7 of Section 4.3 to show from part (b) of Section 4.3 to show that $p$ does not divide $(p-1) !$ and then use Theorem 7 of S ection $4.3 .$ Alternatively, use Wilson's theorem from Exercise $18(\mathrm{b}) . ]$ d) Use part (c) to show that $a^{p} \equiv a(\bmod p)$ for all integers $a .$
Problem [20 points]: Let N = pqr where p, q, r are three distinct primes. Let c ≥ 1 be a positive integer such that c < min{p, q, r}. Suppose x is an integer that satisfies the following conditions: x ≡ c (mod p), x ≡ c (mod q), and x ≡ c (mod r). Prove that x ≡ c (mod N).
Apratim D.
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD