Work modulo 4.
10 = 2 mod 8, 100 = 4 mod 8 and 1000 = 0 mod 8.
Notice an = 1 mod (a - 1).
abcabc = abc · 1001.
Consider the different possibilities modulo 3.
Can you find an inverse for x?
Modular arithmetic.
Extended Euclidean algorithm.
Extended Euclidean algorithm.
Modular arithmetic.
x has an inverse mod m if and only if gcd(x,m) = 1.
Consider the equation mod 9.
Have a look at Exercise 3.
Set up a system of modular equations.
Work mod 10.
Use the chinese remainder theorem
Reduce the problem to an equation with one variable.
Divide both sides by c.
Use the Modular Calculator to factor 2623 and to compute a decoding number. To solve
for w, compute the inverse of 37 mod (p - 1)(q - 1) (using the Modular Calculator for instance).