- 对两个连续整数n和n+1，为什么gcd(n,n+1)=1?
Because when n+1 is divided by n then remainder is 1. Therefore 1 is the GCD of n, n+1
- 利用费马定理计算3^201 mod 11
Fermat theorem explains a^p-1 =1 mod p, where p is a prime number and a is positive number that is not disvisible by p.
3^201 mod 11= (3^1 mod 11)* (3^200 mod 11) mod 11 =(3^1 mod 11) * (3^10 mod 11)^20 mod 11
By using Fermat's rule a^p-1 =1 mod p
= (3^1 mod 11) *(1 mod 11)^20 mod 11
=1* 3 mod 11
=3
- 利用费马定理找一个0~72之间的数a，使得a模73与9^794同余
Fermat's theorem explain a^p-1 = 1 mod p
9^794 mod 73
=(9^720 mod 73) * (9^74 mod 73) mod 73
=(9^72 mod 73)^10 * (9^74 mod 73) mod 73
By fermat's rule 9^72 mod 73=1 mod 73
=1* (9^74 mod 73) mod 73
=(9^72)mod 73 * 9^2 mod 73
=81mod 73
=8
- 利用费马定理找一个位于0~28之间的数 x，使得 x^85 模29与6 同余
Fermat's theorem explains the following,
a^p-1 =1 mod p , p is a prime, a is a positive integer
Another alternative form of Fermat's theorem is used in the problem.
a^p = a mod p, p is a prime, a is a positive