126!, please help,

To find out the number of zero, we can express 126! in form of u x 10^k, where u is an integer not divisible by 10.

10^k = (2^k) (5^k). Therefore, the simplest method would be to express 126! in the form of m x (2^a) x (5^b), where m is a number not divisible by both 2 and 5.

Since 126 >= 2^k gives k <= 6.977, there is no factor of 2^7 = 128 in the expression.

Since 126 >= 5^k' gives k <= 3.005, there is no factor of 5^4 = 625 in the expression.

Therefore, we should consider the number of factors in the expression which are the multiples of 2^6 = 64, 32, 16, 8, ... and also that of 125, 25, 5 in the expression.

Below n is any natural number. floor(x) is the integer part of x.

Number of factors 64n = 1.

Factors 32n includes factors 64n. Thus, no. of factors 32n not divisible by 64 = floor(126/32) - 1 = 3 - 1 = 2

By the same method, no. of factors 16n not divisible by 32 or 64 (e.g. 48) = floor(126/16) - 1 - 2 = 7 - 1 - 2 = 4

No. of factors 8n not divisible by 16 or 32 or 64 (e.g. 24) = floor(126/8) - 1 - 2 - 4 = 15 - 1 - 2 - 4 = 8

No. of factors 4n not divisible by 8 or 16 or 32 or 64 (e.g. 12) = floor(126/4) - 1 - 2 - 4 - 8 = 31 - 15 = 16

No. of factors 2n not divisible by 4 or 8 or 16 or 32 or 64 (e.g. 6) = floor(126/2) - 1 - 2 - 4 - 8 - 16 = 63 - 31 = 32

Thus, a = 6 x 1 + 5 x 2 + 4 x 4 + 3 x 8 + 2 x 16 + 1 x 32 = 120

By using the same approach, we can determine the value of b.

Number of factors 125n = 1.

Factors 25n includes factors 125n. Thus, no. of factors 25n not divisible by 125 = floor(126/25) - 1 = 5 - 1 = 4

By the same method, no. of factors 5n not divisible by 25 or 125 = floor(126/5) - 1 - 4 = 25 - 5 = 20

Thus b = 3 x 1 + 2 x 4 + 1 x 20 = 31

Therefore 126! = m x 2^120 x 5^31 (m is an integer not divisible by both 2 or 5)

Hence, we have 126! = m x 2^120 x 5^31 = m x 2^31 x 5^31 x 2^89 = 2^89 x m x 10^31

Thus, 126! = u x 10^31 with u = 2^89 x m.

Therefore there are 31 zeros.

呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀呀!!!!!!!!!!!!!!!!

經計算，得知126！=
23721732428800468856771473051394170805702085973808045661837377170052497697783313457227249544076486314839447086187187275319400401837013955325179315652376928996065123321190898603130880000000000000000000000000000000
共有52個0。

The answer is
2.37217324288 x 10^211
Checked by www.mathway.com


That means it have 211 zeroes, but also

2 x 10^222

In this case, it got 222 zeroes

Take your pick.

LOL!!

也參考：

https://tw.knowledge.yahoo.com/question/question?qid=1512120903205

https://hk.knowledge.yahoo.com/question/question?qid=7007072904257

https://hk.knowledge.yahoo.com/question/question?qid=7007053104237

2014-06-11 19:26:51 補充：
Chow 恕我直言，你的答案是不正確的。

舉例

1.2345 × 10⁴ = 12345 不會有 4 個零吧。

count number end with 5 and 0

"In the numerical expression of 126!" 應是指下式吧：

1 x 2 x 3 x 4 x ......... x 100 ......... x 123 x 124 x 125 x 126

2014-06-11 14:00:52 補充：
"In the numerical expression of 126!" 應是指下式而不是答案吧？

1 x 2 x 3 x 4 x ......... x 100 ......... x 123 x 124 x 125 x 126

你係咪問最後會出現幾多個零？
中間果 d 零唔計的話，就有 25 + 5 + 1 = 31 個

2014-06-11 16:40:46 補充：
@@. oic. 我英文差 ^^|||
咁慢慢數便可以了