[求助]整除、余数题的解题思路

Springwaters

6、If x and y are integers and xy<>0, what is the remainder when x is divided by y?
(1) when x is divided by 2y, the remainder is 4;
(2) when x+y is divided by y, the remainder is 4;

9、If n=p/q，and both of p and q are non-zero integers, is n an integer?
(1) n^2 is an integer;
(2) n^3 is an integer;

谢谢。

LilyJune

6. (1) X=2Yn+4, Y=3,5,6,7,8,9,10..., X(can be)=10,14,16,18,20,22,24, then the remainders when x is divided by y are 1,4,4,4,4,4,4....that means, if Y&lt;4, then the remainder is 1, if Y&gt;4, then the remainder is 4, so NO. (2) X+Y=Yn+4, X=Y(n-1)+4, Y=3,5,....X=4,9.... the remainders are 1,4,....., so NO if (1)+(2), X=4,14...y=3,5, the remainders are 1,4,.. So, E 9. (1)n^2 is an integer, if n is not an integer, then n must be an irrational number, but n=p/q must be a rational number, so n must be an integer. (2)n^3 is an integer, the same thing. so D. I don't know whether the answers are correct.

shelleyzhdai

6、B

LilyJune

好象6是B，我错了。问一下HZ，如果X=3，Y=4，那X/Y的余数是3吗？X=0*Y+3？

OceanBleu

我也觉得第六题是B
(X+Y)/Y=X/Y+1，那么X/y的余数就是(X+Y)/Y的余数了。至于A，反例很好举

micsunnyfish

能不能讲讲这两题的思路?谢谢.

ZHUGUICHU

6. (1) X=2Yn+4, X/Y=2n+4/Y,当Y&gt;4时，X/Y的余数为4，当Y=4时，X/Y的余数为0，当Y&lt;4时，如等于3，X/Y的余数为1。 (2) X+Y=Yn+4, X=Y(n-1)+4,X/Y=n-1+4/Y，因为由题可知Y&gt;4， 所以X/Y的余数为4。因此，B。 9.(1)n^2是整数，n不是整数就是无理数，由题n=p/q可知n不是无理数，因此n是整数。 (2)同理。因此， D。对这题也没想出其它好的方法。

micsunnyfish

&nbsp;如何从题目知道Y&gt;4?谢谢.

ZHUGUICHU

余数应小于除数。4是余数，Y是除数，因此Y&gt;4。