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標題: [數學,M1&M2] Junior Math [打印本頁]

作者: doraemonserv    時間: 2019-11-8 07:09 AM     標題: Junior Math

Question:

Given x^2 - y^2 = 1001

(a) Prove that x is divisible by 3.
(b) Solve x and y.
作者: doraemonserv    時間: 2019-11-8 07:10 AM

Answer:

x= 75 and y= 68
作者: sealion    時間: 2019-11-8 06:36 PM

仲有X=51,Y=40
同45,32
作者: sealion    時間: 2019-11-8 09:26 PM

Sorry sorry
仲有 501,500
作者: sealion    時間: 2019-11-8 09:28 PM

但係part A 唔知有冇唔用 "暴力法”(即找出所有X的可能)
作者: SC3U    時間: 2019-11-8 10:17 PM

回覆 5#  sealion 的帖子


Try reasoning with modulus.
作者: SC3U    時間: 2019-11-8 10:17 PM

You may also prove that x is odd and y is even
作者: SC3U    時間: 2019-11-9 12:50 AM

Solution for Part (a):

1001MOD3 = 2
Since for all real square n,
引用:
nMOD3 =/= 2


So (y^2)MOD3 = 0 will lead to (x^2)MOD3=2 which is contradiction.

So (y^2)MOD3 = 1 and (x^2)MOD3 = 0

Hence, x^2 must be odd
Since 1001 is odd, y^2 must be even.
作者: SC3U    時間: 2019-11-9 12:52 AM

Solution to Part (b):

(x+y)(x-y) = 1001

Since 1001 = 7 x 11 x 13

You can exhaust all x and y then.
作者: sealion    時間: 2019-11-9 06:43 PM

引用:
原帖由 SC3U 於 2019-11-9 12:52 AM 發表
Solution to Part (b):

(x+y)(x-y) = 1001

Since 1001 = 7 x 11 x 13

You can exhaust all x and y then.


I don't know how use mod
It was out of syllabus at my time.

All I know is to exhaust it. I called it brute force method.

Using mod in (a) is obviously more elegant!

Thank you so much for teaching me!

Maths is so fun




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