请教3题，求过程

crusaderma

In the xy-plane, point (r, s) lies on a circle with center at the origin. What is the value of r2 s2?
(1) The circle has radius 2.
(2) The point (√2, -√2) lies on the circle.

这题错了吧？是不是 最后一句应该是 What is the value of r^2 + s^2?

这题的背景知识，任何一原点为圆心的圆上的任意一点，（x,y）, x^2 +y^2 = 半径的平方，这就是勾股定理

(1) 知道半径了，可以, sufficient
(2) 可以，直接告诉你r 跟 s的值了, sufficient
所以选 D

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If n is a positive integer and r is the remainder when (n-1)(n+1) is divided by 24, what is the value of r?
(1)2 is not a factor of n.
(2)3 is not a factor of n.

(1)
if n = 3, then (n - 1)(n + 1) = 8, so the remainder is 8
if n = 5, then (n - 1)(n + 1) = 24, so the remainder is 0
insufficient

(2)
if n = 2, then (n - 1)(n + 1) = 3, so the remainder is 3
if n = 5, then (n - 1)(n + 1) = 24, so the remainder is 0
insufficient

(together)
the best approach, unless you're really good at number properties, is to try the first few numbers that satisfy both statements, and watch what happens.
if n = 1, then (n - 1)(n + 1) = 0, so the remainder is 0
if n = 5, then (n - 1)(n + 1) = 24, so the remainder is 0
if n = 7, then (n - 1)(n + 1) = 48, so the remainder is 0
if n = 11, then (n - 1)(n + 1) = 120, so the remainder is 0
...you can see where this is headed.

here's the theory:
- if n is not divisible by 2, then n is odd, so both (n - 1) and (n + 1) are even. moreover, since every other even number is a multiple of 4, one of those two factors is a multiple of 4. so the product (n - 1)(n + 1) contains one multiple of 2 and one multiple of 4, so it contains at least 2 x 2 x 2 = three 2's in its prime factorization.
- if n is not divisible by 3, then exactly one of (n - 1) and (n + 1) is divisible by 3, because every third integer is divisible by 3. therefore, the product (n - 1)(n + 1) contains a 3 in its prime factorization.
- thus, the overall prime factorization of (n - 1)(n + 1) contains three 2's and a 3.
- therefore, it is a multiple of 24.
- sufficient

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Last year the price per share of Stock X increased by k percent and the earnings per share of Stock X increased by m percent, where k is greater than m.By what percent did the ratio of price per share to earnings per share increase, in terms of k and m?

这道题说的是，一个公司的股票的股价涨了 K%, 每股收益率涨了m%，问你的是收益率跟每股股价的比率涨了多少

先把条件写出来：
增长之前
股价：x
收益率：y
收益率跟每股股价的比率: x/y
曾涨之后：
增长之前
股价：x（1+ k/100）
收益率：y (1+m/100)
注意一定要处以100，因为给你的k跟m都是%，计算的时候，要转化为小数，所以除以100
收益率跟每股股价的比率: [x（1+ k/100）]/[y (1+m/100)]

收益率跟每股股价的比率 的 增长 [x（1+ k/100）]/[y (1+m/100)] - x/y

问的是比率，所以，综合起来，式子为

{[x（1+ k/100）]/[y (1+m/100)] - x/y} / (x/y) * 100

x/y 全都约掉
成了

{[1+ k/100）]/[(1+m/100)] - 1} / 1 * 100
= [(1+k/100 - 1 - m/100)/(1+m/100)]*100
= [k-m/100+m]*100
= D

tuofujijing1

其实我觉得第二题分析过后，应该可以感觉到出题者的意图：就是把除24简化成为除2，因为除2的特性是其他所数字没有的。如果这里把0也算做余数，那这道题无论最后简化成什么样都没有意义了，可以直接选E了，是不是？

-- by 会员 酸菜鱼315 (2011/10/27 22:32:29)

我后来又想了想 觉得应该是这样的 如果2不是n的因数 那么n=2k+1   (n-1)(n+1)=2k*2(k+1) 因为k(k+1)肯定是2的倍数 那么原式就是8的倍数 所以余数肯定是0 不知道我这么想对不对

酸菜鱼315

是的，其实我一开始的分析是错的。。。不好意思