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2. Find 52019 (mod 13).

2. Find 52019 (mod 13).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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).
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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