作者MysterySW (饭团丸)
看板Grad-ProbAsk
标题Re: [理工] 离散求助!
时间Sun Mar 29 23:00:51 2009
※ 引述《christensen ()》之铭言：
: 1. for every n 属於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为 x + (x^3)/3! + (x^5)/5! + ... = (e^x - e^(-x))/2
1的EGF为 e^x 3的也同
2的为 x + (x^2)/2! + ... = e^x - 1
全部乘起来 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从r=0到无限大
取 (x^20)/20!的系数
所以方法数为 1/2 (4^20 - 2^20)
对吗 我怕我算错@@
: 3. Determine the last digit in (a) 3^55, and (b) 9^1989?
两题都是要求 mod 10 10又跟3 9互质 所以用费马小定理
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小题借用第一小题
9^1989 ≡ 3^3978 ≡ (3^4)^994 * 3^2 ≡ 9 (mod 10)
: 4. if G is a group of order n and a 属於 G, prove that a^n = e.
--
████████ ████████ █ █
◥████████◤ █ █
◥◣ ◢◤ █ █
◥◣ ◢◤ █ █
◥◣◢◤ █ █
◥◤http://www.wretch.cc/album/MysterySW
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 218.166.100.25
※ 编辑: MysterySW 来自: 218.166.100.25 (03/29 23:02)