1.
2003^200
= ((2001 + 2)^2)^100
=(A + 4B + 3 + 1)^100 = (C + 1)^100
这里有50个(C+1)^2相乘,每一个的常数项均为1
其他的带字母项均能被3整除
则余数为1
2.
19^6 = (14+5)^6
=(A + 10B + 21 + 4)^3
=(C+4)^2 * (C+4)
=(D + 14 + 2)(C+4)
= E + 8
= E + 7 + 1
= F + 1
F是7的整数倍,再多一天
那么这天是星期一的后面一天,就是星期二了.
3.
3164^2002
=((3160 + 4)^2)^1001
=(A + B + 15 + 1)^1001
=(C+1)^1001
=(C+1)^1000 * (C+1)
=(C+1)(D+1)
理论同1题,余数也为1.
4.
39271421^6
= (A + 2)^6 = (B + 4)^3
=(B+4)^2 * (B+4)
=(C + 9 + 7)(B +4)
= D + 28 = D + 27 + 1
余数为1
5,6,7,8均可采用此类方法
9.
1^1 + 2^2 + 3^3 + 4^4 + 5^5 + 6^6 + 7^7 +8^8 +9^9
=1 + (3 + 1) + 3^3 + (3+1)^4 + (3+2)^5 + 6^6 + (6+1)^7 + (6 + 2)^8 + 9^9
= 1 + 1 + 0 + 1 + A +(3+2)^5 + (6+1)^7 + (6 + 2)^8
= B + (3+2)^5 + (6+1)^7 + (6 + 2)^8
后面解法同上
10.
(7^7)^7
=7^49
= (5 + 2)^48 * 7
= (A + 4)^24 *7
= (B + 15 + 1)^12 *7
= (C + 1) ^12 * 7
= (D + 1 ) *7
= E + 7 = E + 5 + 2
余数为2
如果此批题有疑问,欢迎追问或者另行开题追答.