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4^(n + 1) + x and 4^(2 n) – x are divisible by 5

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4^(n + 1) + x and 4^(2 n) – x are divisible by 5

by sanju09 » Tue Oct 12, 2010 5:55 am
For what least value of x, both 4^(n + 1) + x and 4^(2 n) - x are divisible by 5, n is an even positive integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


[spoiler]Made up[/spoiler]
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by deepaks04 » Tue Oct 12, 2010 6:17 am
For n=2, minimum value of x=1

number having unit digit in 0 or 5 are divisible by 5

No power raise to 4 will give 0 as unit digit, so option A is ruled out

Ans. B) 1
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by sanju09 » Tue Oct 12, 2010 10:17 pm
deepaks04 wrote:For n=2, minimum value of x=1

number having unit digit in 0 or 5 are divisible by 5

No power raise to 4 will give 0 as unit digit, so option A is ruled out

Ans. B) 1
B is right, but a better explanation is still awaited.
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by gmatmachoman » Wed Oct 13, 2010 11:39 am
sanju09 wrote:For what least value of x, both 4^(n + 1) + x and 4^(2 n) - x are divisible by 5, n is an even positive integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


[spoiler]Made up[/spoiler]
4^(n + 1) will always end with a 4 in its unit digits provided n is an even positive integer. So we need X to be 1 so that

4^(n + 1) + x is divisible by 5

4^(2 n) always end with 6 in its unit's digits provided n is an even positive integer. So we need X to be 1 so that

4^(2 n) - x is divisible by 5.

But Sanju, only 1 can satisfy the given condition.. why all other options?? Not sure whether i was rite! feeling sleepy

I think u can modify the options like 1,6,11,16...
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by fskilnik@GMATH » Wed Oct 13, 2010 1:28 pm
sanju09 wrote:For what least value of x, both 4^(n + 1) + x and 4^(2 n) - x are divisible by 5, n is an even positive integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Hi sanju09,

Nice idea for a problem, and I guess gmatmachoman "got the picture" and realized that the "key" was controlling the unit´s digit of the integer powers of 4...

Just a small detail: I guess you should have put that x is (for instance) a non-negative integer in the question stem, otherwise it is not hard to see that x=-4 is also possible, and also x=-4-5=-9 , x=-14, etc...
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by sanju09 » Wed Oct 13, 2010 9:41 pm
fskilnik wrote:
sanju09 wrote:For what least value of x, both 4^(n + 1) + x and 4^(2 n) - x are divisible by 5, n is an even positive integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Hi sanju09,

Nice idea for a problem, and I guess gmatmachoman "got the picture" and realized that the "key" was controlling the unit´s digit of the integer powers of 4...

Just a small detail: I guess you should have put that x is (for instance) a non-negative integer in the question stem, otherwise it is not hard to see that x=-4 is also possible, and also x=-4-5=-9 , x=-14, etc...
Thank you very much fskilnik for liking the idea, yeah gmatmachoman did justice with that and he came up with a nice suggestion too. Although, we can ignore his this statement that "4^(2 n) always end with 6 in its unit's digits provided n is an even positive integer", but we cannot ignore his suggestion to modify answers, really; let's not forget that he was feeling sleepy too.

I must confess in the end that the question should have been worded as under:

For what least non-negative integer value of x, both 4^(n + 1) + x and 4^(2 n) - x are divisible by 5, n is an even positive integer?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4


Thanks
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by fskilnik@GMATH » Tue Oct 26, 2010 9:03 am
sanju09 wrote:Thanks
My pleasure! (Sorry for this delay; I did not see you reply till now!!)
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by goyalsau » Wed Oct 27, 2010 1:08 am
fskilnik wrote:
Just a small detail: I guess you should have put that x is (for instance) a non-negative integer in the question stem, otherwise it is not hard to see that x=-4 is also possible, and also x=-4-5=-9 , x=-14, etc...
I must say you have a excellent approach, Its been pleasure reading all your comments and suggestions.....

Every now and then you prove the point that are very critical..
Thanks a lot
Saurabh Goyal
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by fskilnik@GMATH » Wed Oct 27, 2010 1:20 am
goyalsau wrote:
fskilnik wrote:
Just a small detail: I guess you should have put that x is (for instance) a non-negative integer in the question stem, otherwise it is not hard to see that x=-4 is also possible, and also x=-4-5=-9 , x=-14, etc...
I must say you have a excellent approach, Its been pleasure reading all your comments and suggestions.....

Every now and then you prove the point that are very critical..
Thanks a lot
Thank YOU for your kind words, goyalsau!

All the best,
Fabio.
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