2. Find 52019 (mod 13).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Use Chinese Remainder Theorem and Fermat's Little Theorem
![To handle fast exponentiation, we use CRT along with Fermat's Little Theorem:
Theorem 1 (Fermat's Little Theorem (FLT)). If p is prime and a is an integer not
divisible by p, then
ap- = 1 (mod p).
Furthermore, for every integer a we have
a³ = a (mod p).
The use of FLT is that we can compute exponentiation in modular arithmetic quickly.
Example 1. Find 7222 (mod 11).
Note that 710 = 1 (mod 11) by FLT. Pulling out 710 from 7222 is the same as pulling
out 1 (mod 11) so
7222710-22+2 = 710-22.72 = (710) 22. 72 = 122.7² = 7² = 5 (mod 11).
2. Find 52019 (mod 13).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90d88307-63d1-48ae-b5f3-220bf176d5b0%2F72a8b0de-58de-4ebe-ae19-a232c7538941%2F533v39_processed.png&w=3840&q=75)
Transcribed Image Text:To handle fast exponentiation, we use CRT along with Fermat's Little Theorem:
Theorem 1 (Fermat's Little Theorem (FLT)). If p is prime and a is an integer not
divisible by p, then
ap- = 1 (mod p).
Furthermore, for every integer a we have
a³ = a (mod p).
The use of FLT is that we can compute exponentiation in modular arithmetic quickly.
Example 1. Find 7222 (mod 11).
Note that 710 = 1 (mod 11) by FLT. Pulling out 710 from 7222 is the same as pulling
out 1 (mod 11) so
7222710-22+2 = 710-22.72 = (710) 22. 72 = 122.7² = 7² = 5 (mod 11).
2. Find 52019 (mod 13).
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