666..6 * 666…67 ( 乘数有2002个6 ,被乘数有2002个6,1个7)
1个回答
答案为:44...422..22(2002个4与2002个2)
(注:下解中^为乘方号,例如10^k表示10的k次方)
归纳法:(1)6*7=42,6+7=13 成立.
66*67=4422,66+67=133成立
(2)设k位时,有:
66...6(k位)*66...67(k位)=44...422..2(k个4与k个2).
66...6(k位)+66...67(k位)=133...3(k个3)
则当k+1位时:设x=66...6(k位),y=66...67(k位),
x*y=44...422..2(k个4与k个2).
x+y=133...3(k个3)
66...6(k+1位)*66...67(k+1位)=[6*(10^k)+x]*[6*(10^k)+y]
=6*(10^k)*[6*(10^k)+x+y]+x*y=6*733...3*(10^k)+44...422..2(k个3,k个4与k个2)
=4399..98*(10^k)+44...422..2(k-1个9,k个4与k个2)
=44...422..2(k+1个4与k+1个2)
(3)由上证明,得:66...6(n位)*66...67(n位)=44...422..2(n个4与n个2).
(4)当n为2002时,就是答案.
