(2+1)(2^2+1)(2^4+1).(2^512+1)的末尾数字是多少

题目

(2+1)(2^2+1)(2^4+1).(2^512+1)的末尾数字是多少

答案

(2+1)(2^2+1)(2^4+1)...(2^512+1)
=(2-1)(2+1)(2^2+1)(2^4+1)...(2^512+1)
=(2^2-1)(2^2+1).(2^512+1)
=...
=(2^512-1)(2^512+1)
=(2^1024-1)
2^1=2
2^2=4
2^3=8
2^4个位数是6
2^5个位数是2
所以个位数是4个一循环
1024/4=256
所以2^1024的个位数=2^4的个位数=6
即：2^1024-1的个位上是：6-1=5