Show that 2269^1997 + 1779^1997 + 1730^1997 - 1776^1997 is divisible by 2001.?

In fact it is true for any odd "n" i.e. if n is an odd positive integer, then
2269^n + 1779^n + 1730^n - 1776^n
is an integer multiple of 2001

For answer have a look at following page:
http://mathforum.org/library/drmath/view/51589.html

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Nino_ops: It can not be any power, "it has to be odd power" check it out yourself for n=2,4,6 etc

I do not know about what the thumb downers think but it's really shocking that they seem to miss such a simple fact!

Power of 1997 isn't important
it could be any power

and to proove for 2001
prove the function is divisible by
3 , 23 and 29 (2001 = 3 x 23 x 29)

little bit hard to prove for 23 and 29
but u can continue from here solve it for 3 23 29

As the numbers given are all to the same power just add the numbers to a power one because if the solution is divisible by 2001 then any power of the final number will be divisible by 2001

So 2269 +1779 + 1730 - 1776 = 4002

Divided by 2001 = 2

You can see that if 4002 was squared or cubed and so on it would factorise by 2001


e.g.
If you wer adding 4x^2 + 8x^2 you would see straight away that it was 12x^2

we first factoise 2001 = 3 * 23 * 29

as 3 , 23, and 29 are pimes so if above is divisible by 3 23 and 29 then it is dvisible by 2001

now a^n-b^n ( n odd or even) is divisble by (a-b)

and a^n + b^ n is divisible by a+ b when b is odd

we group the sum pair wise to show divisiblity by each number

2269^1997 + 1730^1997 is divisible by 3999 hence 3
as 3999 is divisible by 3
and

1779^1997-1776^1997 is divisible by 1779-1776 or 3

1776-1730 = 46 = 2* 43

so 1776^1997 - 1730^ 1997 is divisible by 23

2269^1997+ 1779^1997 is divsible by 2269+ 1779 = 4048 or 23* 176

so it is divisible by 23

now for 29

2269 - 1776 = 493 = 29* 17

and 1779+1730 = 3509 = 29* 121

so 2269^1997-1776^1997 is divisible by 29 and 1779^1997 + 1730^1997 is divsible by 29

adding them we see that above expression is divisible by 3 23 and 29 and hence their product is 2001

Do you have any specific complaint about Bush or just trite and general whining? Maybe we should have just wrung our hands after 9-11, I'm sure the terrorists would have taken that as a sign to leave us alone. After all, since Spain ran away with their tail between their legs, they haven't attacked them. Oh wait, yes they did. Same with France, no matter how much comfort and cover they provide, they're still targets for these crazies. We certainly have ruined the environment under Bush, right? No, wait, I guess not. Cleaner water and air, still no drilling in ANWR, he's even caved on global warming and agreed to a program parallel to Kyoto without surrendering our sovereignty to the UN. Yes, he's a disaster as president. Wouldn't it be much better to have our troops out of Iraq so these jihadists could be attacking schools and shopping malls here in the US? Definitely better not to meet them head on with well-trained soldiers, let mall rent-a-cops deter that threat. I like the Onion but I don't see anything prophetic about this. I seem to recall hearing that Clinton deployed our troops more than once, even putting them under UN commanders which I hope we never do again. They swear an oath to our country so let's not put them in a position where that loyalty might be put in jeopardy.