數學有問題..請教..10分..THX..步驟..^^

Peter selects those which are squares or cubes among the first 2003 positive integers and then adds them up. What is the unit digit of his result?
(0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225,256,289,324,361,400,441,484,529,576,625,676,729,784,841,900,961,1024,1089,1156,1225,1296,1369,1444,1521,1600,1681,1764,1849, 1936).....2003之內平方數
0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728).......2003之內立方數
相加之後是35454
unit dight 是4
2.Find the smallest positive multiple of 2007 which has exactly 2007 positive factors.
2007的factors有：1, 3, 9, 223, 669, 2007
=1x2007
=2007

2007-04-10 17:46:41 補充：
我個是蠢方法，你應該以網友(jh1090_2005 @ 收拾心情，準備上學
)的方法去思考才行....^^

1.

For the squares :

sequence :

1 , 4 , 9 , 6 , 5 , 6 , 9 , 4 , 1 , 0 , 1 , 4 , 9 , 6 , 5 , 6 , 9 , 4 , 1 , 0 , 1 , ...

The largest square < 2003 is

44 ^2 = 1936

so the sum of the squares are:

(1 + 4 + 9 + 6 + 5 + 6 + 9 + 4 + 1 + 0) x 4 + 4

= 45 x 4 +4

= 184

For the cubes ,

sequence :

1 , 8 , 7 , 4 , 5 , 6 , 3 , 2 , 9 , 0 , 1 , 8 , 7 , 4 , 5 , 6 ,3 , 2,9 , 0, 1 ...
the largest cube < 2003 is
12^3 = 1728
so the sum of the cubes are
1 + 8 + 7 + 4 + 5 + 6 + 3 + 2 + 9 + 0 + 1 + 8
= 54

so the unit digit of his result
= 54 + 184
= 238
the unit digit of his result is 8

2.
The smallest positive multiple of 2007 which has exactly 2007 positive factors
should be
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x ... x 2005 x 2006 x 2007
= 2007!