小學奧數數學題_2

1 x 2 x 3 x 4 x 5 x 6 x ----------- x 26

= 26!

= 403291461126605635584000000

最後有8個0呀!!!

10=2*5 每一個『０』必須有一個2和一個5

1-10 has2*5*10=2個『０』
11-20 has12*(3*5)*20=2個『０』
21-26 has 24*25=(2*2*6)*(5*5)=2個『０』

2+2+2=6

So the answer is 6個『０』

Prime factorization, consider 2, 3, 5, 7, 11, 13, 17, 19, 23

how many 23? 1
19: 1
17: 1
13: 26/13 = 2
11: 26/11 = 2
7: 26/7 = 3

5:
we need to consider 25 first
25: 26/25 = 1
5: 26/5 - 1 = 5 - 1 = 4

so the power of 3 is 4 + 2*1 = 6

3:
we need to consider 9 first
9: 26/9 = 2
3: 26/3 - 2 = 8-2 = 6

so the power of 3 is 6 + 2*2 = 10

2:
we need to consider 16, 8, 4, 2...
16: 26/16 = 1
8: 26/8 - 1 = 3-1 = 2
4: 26/4 - (1+2) = 6-3 = 3
2: 26/2 - (1+2+3) = 13-6 = 7

so the power of 2 is 7 + 2*3 + 3*2 + 4*1 = 23

i.e. 1 x 2 x 3 x 4 x 5 x 6 x ... x 26
= 223 x 310 x 56 x 73 x 112 x 132 x 17 x 19 x 23

we have the factor of 26 x 56
so there are 6 '0's at the end

2007-04-11 12:45:24 補充：
for the purpose of this question, we do not need to consider other prime factors (i.e. 3, 7, 11, 13, 17, 19, 23)i just try to give a more complete picture of the story behind

2007-04-11 14:09:14 補充：
for the answer about 1-10, 11-20, 21-26....what if the product of a number from 1-10 and another number from 21-26 contribute to a multiple of 10?

2007-04-21 04:16:40 補充：
你o係比賽會唔會計到26!呀...
其實我都講o左, 你只要睇有幾多個2幾多個5就得喇...
有咩email我嘛...