[求助]请问费费第四部分35题能帮忙解释一下求证过程吗？

35、N是正整数，N是否为一个整数的平方？
（1）4N是一个整数的平方；
（2）N^3是一个整数的平方；
答案是D。仔细一较真还真搞不懂。谢谢！

i am not NN, but let me try.

from condition 1, N is integer greater than 0, let N =1, then 4*1 = 4, which is 2^2; let N = 2, then 4*2 = 8, which is not satisfied the condition; but, when N = 4, 9, 16, 25, ....., 4*N is actually the square of an integer, because 4 by itself is 2^2, when it times square of any integer, the result is always the square of an integer. (hope you understand my explanation for condition 1.)

from condition 2, N^3 is an integer's square. From question, we know that N is integer greater than 0; therefore, let N =1, then N^3 = 1, which N is the square of integer 1; let N = 4, then N^3 = 64, which N is the square of integer 2; let N = 9, then N^3 = 729, which N is the square of integer 3; so on and so forth, whenever N equals to the square of any positive integer, it fulfills the condition.

so lengthy, hope you could understand.

楼上的只举了几个可以成立的例子，但是并不等于穷举出来所有的可能性，

还是不太明白，N^3是k^2与N是平方数的必然关系在哪里，请指点一下。

在求证N是平方数，你把它当成已知条件了；怎么能保证N如果不是平方数，你的推论就不成立呢？

N^3=K^2 => N*N^1/2=K =〉N=(K/N)^2

我知道了，不用推到最后，只要到N*N^1/2=k就可以了，因为N，K都是整数，因此，N^1/2一定是整数，因此N一定是平方数。

忽然发现当时自己钻了牛角尖，其实直接从N^2*N=k^2就可以推倒出来了。

谢谢！

哎呀，我总算明白了

1）4N=K^2  -->  N= (k/2)^2, 因为N是整数，所以k/2也是整数，所以N是整数的平方；

2）N^3=K^2 -->  N= (k/N)^2, 因为N是整数，所以k/N也是整数，所以N是整数的平方。