作者MysterySW (飯糰丸)
看板Grad-ProbAsk
標題Re: [理工] 離散求助!
時間Sun Mar 29 23:00:51 2009
※ 引述《christensen ()》之銘言：
: 1. for every n 屬於Z^+, n>=14, prove that n can be written as a sum of
: summands of 3's and/or 8's.
: 2. how many 20-digit quaternary (0,1,2,3)sequences are there, where:
: a) there is at least one 2 and an odd number of 0's ?
0的EGF為 x + (x^3)/3! + (x^5)/5! + ... = (e^x - e^(-x))/2
1的EGF為 e^x 3的也同
2的為 x + (x^2)/2! + ... = e^x - 1
全部乘起來 1/2 [e^4x - e^(-2x) - e^3x + e^(-3x)]
= 1/2 { E[(4x)^r]/r! - E[(-2x)^r]/r! - E[(3x)^r]/r! + E[(-3x)^r]/r! }
summation從r=0到無限大
取 (x^20)/20!的係數
所以方法數為 1/2 (4^20 - 2^20)
對嗎 我怕我算錯@@
: 3. Determine the last digit in (a) 3^55, and (b) 9^1989?
兩題都是要求 mod 10 10又跟3 9互質 所以用費馬小定理
3^Φ(10) ≡ 1 (mod 10) Φ(10) = 4
所以 3^4 ≡ 1 (mod10)
又 55/4 = 13 ... 3
3^55 ≡ (3^4)^13 * 3^3 ≡ 7 (mod 10)
b小題借用第一小題
9^1989 ≡ 3^3978 ≡ (3^4)^994 * 3^2 ≡ 9 (mod 10)
: 4. if G is a group of order n and a 屬於 G, prove that a^n = e.
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※ 編輯: MysterySW 來自: 218.166.100.25 (03/29 23:02)