设a.b,c,d∈N,证明(a的(4b+d)次方)减去（a的（4c+d）次方）能被240整除a4b+d-a4c+d

设a.b,c,d∈N,证明(a的(4b+d)次方)减去（a的（4c+d）次方）能被240整除a4b+d-a4c+d
设a.b,c,d∈N,证明(a的(4b+d)次方)减去（a的（4c+d）次方）能被240整除
a4b+d-a4c+d

设a.b,c,d∈N,证明(a的(4b+d)次方)减去（a的（4c+d）次方）能被240整除a4b+d-a4c+d
240=3*5*16,3,5,16两两互质.
只要证明
A=a^(4b+d)-a^(4c+d)分别被3,5,16整除即可.
如果你知道费马小定理那就很容易了.否则需要用带余数除法慢慢整.怕你不知道费马小定理,就慢慢来吧.
预备知识：
1.3不整除x,那么3整除x^2-1,或者说x^2除以3余1.
这是因为x^2-1=(x-1)(x+1),x除以3的余数是1的话,x-1是3的倍数,x除以3的余数是2的话,x+1是3的倍数,总之,x^2-1是3的倍数.
2.5不整除x,那么5整除x^4-1,或者说x^4除以5余1.
这是因为
x^4-1=(x^2-1)(x^2+1)=(x^2-1)(x^2-4+5)
=(x-2)(x-1)(x+1)(x+2)+5(x^2-1)
无论x除以5余1,2,3,4上式都是5的倍数.
3.偶数的4次方是16的倍数,奇数的平方被8除余1,奇数的4次方被16除余1.
第一句话显然正确,第三句话可由第二句推出.
只证明：奇数的平方被8除余1.
因为(2k+1)^2=4k(k+1)+1,k和k+1一定有一个偶数,所以4k(k+1)是8的倍数.
有了这些准备,就很容易了.
看式子 A=a^(4b+d)-a^(4c+d)=a^d[(a^b)^4-(a^c)^4]
=a^d[(a^4)^b-(a^4)^c]
如果a是2,3,5的倍数,显然A就是16,3,5的倍数.下面设a不是2,3,5的倍数.
3不整除a,根据预备知识1（a^b)^4和(a^c)^4除以3都余1,二者之差就是3的倍数.5不整除a,(a^b)^4和(a^c)^4除以5都余1,二者之差是5的倍数.2不整除a,
(a^b)^4和(a^c)^4除以16都余1,二者之差是16的倍数.
证毕.

This past May, our own Lovely Kicks shop held a vault release of the Air Jordan 10 “Steel Grey.” The event was a great success & lots of had the chance to buy a pair of these classic Jorda...

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This past May, our own Lovely Kicks shop held a vault release of the Air Jordan 10 “Steel Grey.” The event was a great success & lots of had the chance to buy a pair of these classic Jordans from 2005. Carrying on with the success of the colorway & design, Jordan Brand has decided to generate a fusion version.
The Air Jordan Fusion 10 “Steel Grey” features an upper that is similar to the original Air Jordan 10. With a white leather upper, black & grey striped lace eyelets,cheap nike shox, as well as a black tongue, this pair of fusions retains a simple yet stylish base. Changes to this classic Jordan footwear includes the use of an Air Force 1 outsole, midsole, & strap, which encircles the upper ankle. Still, various Jordan stats appear at the bottom of the outsole, an attribute that distinguishes the Air Jordan 10 from the rest. There is no word yet on a retail date; however, is this Fusion better than some of the other Air Jordan Fusions we’ve seen over the past couple of years?

Air Jordan Fusion 10 “Steel Grey”

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因为 a的4次方为偶数 so(a的(4b)次方)减去（a的（4c）次方）依然为偶数 若d为奇数 a也为奇数 则(a的(4b+d)次方) 也为奇数 （a的（4c+d）次方） 也为奇数 而奇数减奇数为偶数 又因为a为偶数时 (a的(4b+d)次方)减去（a的（4c+d）次方）定为偶数 注【题目 a.b,c,d∈N,应为整数集】 若符合整数及则 定能被240整除...

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因为 a的4次方为偶数 so(a的(4b)次方)减去（a的（4c）次方）依然为偶数 若d为奇数 a也为奇数 则(a的(4b+d)次方) 也为奇数 （a的（4c+d）次方） 也为奇数 而奇数减奇数为偶数 又因为a为偶数时 (a的(4b+d)次方)减去（a的（4c+d）次方）定为偶数 注【题目 a.b,c,d∈N,应为整数集】 若符合整数及则 定能被240整除

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