Question

一道国外的大1数学题，帮帮忙谢谢

求救啊，刚到国外。代数课课老师布置了些作业，这道题都蒙圈了，完全不会做。麻烦帮帮忙吧，谢谢了

Answer 1

(a) the product (called P) has 2 as one of it's factors, so P is also even according to definition.

(b)
Not sure what's asked. Assume it asked to prove 3 is always a factor of n(2n+1)(4n+1):
(i) n = 3k (n, k are integers)
It's clear, 3 is a factor of n(2n + 1)(4n + 1)
(ii) n = 3k + 1

2n + 1 = 6k + 3 = 3(2k + 1), 3 is a factor
(iii) n = 3k + 2
4n + 1 = 12k + 9 = 3(4k + 3), 3 is a factor

(3)

Suppose √3 is a rational number, then it can be expressed in the form of a/b, here a and b are both integers and the only common factor is 1.
√3 = a/b
3 = a²/b²
a² = 3b² (1)

3 is a factor of 3b², so a² must has 3 as a foctor, let a = 3c
(3c)² = 3b²
b² = 3c²
Follow the same reasoning, b has 3 as a factor; so a and b have 1 and 3 as common factors. This contradicts the initial assumption, so √3 is not a rational number.

追问

不好意思啊，能麻烦您把这道也看看么，谢谢了

追答

(lna)(lnb)/lnc = (lna)(lnb)/lnc

[(lnb)/lnc]lna = [(lna)/lnc]lnb

Answer 2

说中文

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