Question

离散数学

What are the solutions of the linear congruence 3x = 4 (mod 7)?

Solution: By Example 3 we know that -2 is an inverse of 3 modulo 7. Multiplying both sides
of the congruence by - 2 shows that
-2*3x = -2*4 (mod 7).
Because -6 = 1 (mod 7) and -8 = 6 (mod 7), it follows that if x is a solution, then x = -8 =6 (mod 7).
We need to determine whether every x with x = 6 (mod 7) is a solution. Assume that
x = 6 (mod 7). Then, by Theorem 5 of Section 3.4, it follows that
3x = 3 *6 = 18 = 4 (mod 7),
这句
Because -6 = 1 (mod 7) and -8 = 6 (mod 7), it follows that if x is a solution, then x = -8 =
6 (mod 7).

是根据什么得出的结论



Answer 1

它是用二元运算的性质求解的，3在模7下乘法下的逆元是-2，然后-8＝6(mod 7)，得x=6。
－－－－
我是这样做的：正整数集{1,2,3,4,5,6}关于模7乘法构成群，1是单位元，3的逆元是5。
所以由3x=4(mod 7)，左乘5，得x=20(mod 7)，20=6(mod 7)，所以x=6。
－－－－
也可以这样做，3x=4(mod 7)，则3x-4被7整除，最小的正整数x即为6

追问

二元运算的那个性质？

追答

群中，运算满足消去律

Answer 2

不会