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cs55555

Joined: 23 Aug 2004Posts: 19

Posted: Mon Aug 23, 2004 7:57 pm    Post subject: 計和~~~唔該幫幫手~

設 的各位數字之和為 A , 而 A 的各位數字之和為 B , 求 B 的各位數字之和 C

請用中文詳盡地解釋~

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Carto

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Joined: 29 Oct 2003Posts: 322

Location: Hong Kong

Posted: Wed Aug 25, 2004 7:50 pm    Post subject: Re: 計和~~~唔該幫幫手~

cs55555 wrote:

設 的各位數字之和為 A , 而 A 的各位數字之和為 B , 求 B 的各位數字之和 C

請用中文詳盡地解釋~

解這題的關鍵是發現到 C 的數值其實很小。由於 , 在十進制中的位數是 16211,所以 。A 的數值已經比原來的 細了很多。再取數字和,,已經變成兩個位以內的數。再取一次數字和,不難知道。另一方面,由於 ,所以 C = 7。_________________

世上沒有完美的人完美的事,而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來,與時代一起進步去追求無止境的完美。

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cs55555

Joined: 23 Aug 2004Posts: 19

Posted: Thu Aug 26, 2004 11:11 am    Post subject: 計和~~~唔該幫幫手~

Quote:

解這題的關鍵是發現到 C 的數值其實很小。由於 , 在十進制中的位數是 16211,所以 。A 的數值已經比原來的 細了很多。再取數字和,,已經變成兩個位以內的數。再取一次數字和,不難知道。另一方面,由於 ,所以 C = 7。

我唔明點解會得出,


這幾步~

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Carto

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Joined: 29 Oct 2003Posts: 322

Location: Hong Kong

Posted: Thu Aug 26, 2004 6:31 pm    Post subject:

Quote:

我唔明點解會得出,

這幾步~

因為 在十進制中有 16211 個位,每個位都係 0 至 9,所以 的數字和不會大於 的數字和,即是說 。_________________

世上沒有完美的人完美的事,而我們的責任就是要令自己的表演達至最精彩最完美。所以魔術一直都沒有停頓下來,與時代一起進步去追求無止境的完美。

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Milton

Frequent VisitorJoined: 27 Oct 2003Posts: 242

Location: HKUST Math

Posted: Thu Aug 26, 2004 10:33 pm    Post subject:

Just a note to help cs55555 understand.

We consider because it can tell us the number of digits of an integer in decimal form. Like in this question, we have . Hence has actually 16210 digits. Finally, to see the conclusion made by Carto, note that considering an integer mod 9 is equivalent to considering the sum of its digits mod 9.

[Edited by Andy. The LaTeX content 4444^4444 was changed to 4444^{4444} so that the exponent 4444 is correctly displayed as it should be. If you think this change is inappropriate, please feel free to tell me. Thanks.]

_________________Sometimes Truth is meanlingless;What means is how you believe in.

偶爾,真相並沒有意義;意義在於你怎樣相信。

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cs55555

Joined: 23 Aug 2004Posts: 19

Posted: Fri Aug 27, 2004 9:51 am    Post subject:

thanks

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Curtiss20184

Frequent VisitorJoined: 30 Aug 2004Posts: 38

Location: HK F7

Posted: Mon Aug 30, 2004 4:09 pm    Post subject:

我都係睇唔明呀...

咁點解C 唔係12 而係 7 既….

我唔明最後mod 9 用黎做乜野呀…

_________________

I am a Form 7 Student in Hong Kong.

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Peter

Frequent VisitorJoined: 18 Jan 2004Posts: 128

Location: Hong Kong

Posted: Mon Aug 30, 2004 4:44 pm    Post subject:

Curtiss20184 wrote:

我都係睇唔明呀...

咁點解C 唔係12 而係 7 既….

我唔明最後mod 9 用黎做乜野呀…

正如Milton所說,

Milton wrote:

note that considering an integer mod 9 is equivalent to considering the sum of its digits mod 9.

任何一個正整數均可唯一寫成,其中()為小於10的非負整數,且。
換言之,的十進制表示是。(例如。)
可以證明,被9除時所得的餘數,與的數位和()被9除時所得的餘數相等。 所以雖然經過多次轉換(求數字和),但是轉換後的數被9除時所得的餘數恆不改變。

由於,且被9除時餘數為7,所以必為7。

_________________

Therefore do not worry about tomorrow, for tomorrow will worry about itself. Each day has enough trouble of its own.

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Curtiss20184

Frequent VisitorJoined: 30 Aug 2004Posts: 38

Location: HK F7

Posted: Mon Aug 30, 2004 7:17 pm    Post subject:

明白啦..
但係點知 4+4+4+4 mod 9 = 7 4444次方唔駛理~?_________________

I am a Form 7 Student in Hong Kong.

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Kenny TM~

Frequent VisitorJoined: 20 Jan 2004Posts: 141

Posted: Mon Aug 30, 2004 7:28 pm    Post subject:

Curtiss20184 wrote:

明白啦..
但係點知 4+4+4+4 mod 9 = 7

4444次方唔駛理~?

~

(Problem: Prove ?)

(*Maybe* n-1 should be replaced with ; I haven’t investigated this very much)

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Siutsz

Frequent VisitorJoined: 25 Dec 2003Posts: 37

Location: HK

Posted: Mon Aug 30, 2004 8:27 pm    Post subject:

it should be
also, when u use Euler Thm, u should check that , or equivalently

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Curtiss20184

Frequent VisitorJoined: 30 Aug 2004Posts: 38

Location: HK F7

Posted: Tue Aug 31, 2004 1:41 am    Post subject:

OH thanks a lot 但我未學過.. pure會唔會教架~~??_________________

I am a Form 7 Student in Hong Kong.

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Kenny TM~

Frequent VisitorJoined: 20 Jan 2004Posts: 141

Posted: Tue Aug 31, 2004 11:13 am    Post subject:

Curtiss20184 wrote:

OH thanks a lot 但我未學過..

pure會唔會教架~~??

根據2005年個課程: 唔會. 香港D中學課程好似除左 Prime factorization 同 GCD (=HCF)、LCM 之外冇乜講 number theory…

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Curtiss20184

Frequent VisitorJoined: 30 Aug 2004Posts: 38

Location: HK F7

Posted: Tue Aug 31, 2004 12:02 pm    Post subject:

OH i see le… 我algebra好過calculus好多.. calculus d locus搞到我好亂…. algebra差complex number同埋limit未教… 所以我都做左d past paper嚕~

但係paper 2 一份都未操過~

_________________

I am a Form 7 Student in Hong Kong.

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billyauhk

Joined: 21 Mar 2005Posts: 9

Posted: Mon Mar 21, 2005 5:08 pm    Post subject:

我連卑個mod字班中六都唔識破!!!_________________

A strange F.3 student

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