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What is the remainder when ((55)^15!)^188 is divided by 17

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What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 10:10
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What is the remainder when ((55)^15!)^188 is divided by 17?

A) 1
B) 16
C) 3
D) 5
E) 15


PS: I am not sure whether this falls under 600-700 difficulty level.
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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 10:23
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What is the remainder when ((55)^15!)^188 is divided by 17?
A) 1
B) 16
C) 3
D) 5
E) 15

((55)^15!)^188

When 55 is divided by 17, the remainder is 4
because we are interested in remainder, we can reduce the previous question as(4^15!)^188

15! will be a even number, so we can safely divide 15! by 2 & raise the power of 4 by 2 i.e.

={16^(15!/2)}^188
={[17-1]^(15!/2)}^188
={[-1]^(15!/2)}^188

Negative 1 raised to any even power will reduce to +1

={1}^188
=1

Thus the remainder is 1

So the answer is A.

Hope this explanation helps.

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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 13:33
Why did you plug the remainder of 4 into the problem? I understand 55/17 gives remainder of 4 and I see where you went with the problem but I don't understand why you put the remainder back into the problem? Im lost!
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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 14:46
When a number a is divided by another number b, the quotient would be some number q and we would get a remainder r. Since we need to find the remainder, we can ignore the quotient and feed in the remainder. On feeding in the remainder and raising it to the power, we can find the total remainder.

Hope this helps! Let me know if you need any further notification.

Richard0715 wrote:
Why did you plug the remainder of 4 into the problem? I understand 55/17 gives remainder of 4 and I see where you went with the problem but I don't understand why you put the remainder back into the problem? Im lost!

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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 17:21
Thanks Kris, I appreciated it! I am just trying to apply the concept to something else but how would I go about this problem? What is the remainder when ((17)^10)^123) is divided by 3? 17/3 has a remainder of 2, feeding in 2 into the problem how would I then find the remainder? Thanks again!





Kris01 wrote:
When a number a is divided by another number b, the quotient would be some number q and we would get a remainder r. Since we need to find the remainder, we can ignore the quotient and feed in the remainder. On feeding in the remainder and raising it to the power, we can find the total remainder.

Hope this helps! Let me know if you need any further notification.

Richard0715 wrote:
Why did you plug the remainder of 4 into the problem? I understand 55/17 gives remainder of 4 and I see where you went with the problem but I don't understand why you put the remainder back into the problem? Im lost!
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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 23 Feb 2013, 21:54
We need to find the remainder when ((17)^10)^123) is divided by 3
As you suggested let us divide, 17/3 and feed in the remainder back into the question.

((2)^10)^123)= (1024)^123/3
You can divide 1024 by 3. The remainder is 1. Hence 1^123=1

This was possible because the remainder of 17/3 was a small number,2

Another way of doing this is

((2)^10)^123)=((3-1)^10)^123
When you divide the data inside the bracket by 3, 3 would get completely divided by 3 and hence can be ignored.

((-1)^10)^123
Any negative number raised to an even integer= positive number

1^123=1

Hope that helps!

[quote="Richard0715"]Thanks Kris, I appreciated it! I am just trying to apply the concept to something else but how would I go about this problem? What is the remainder when ((17)^10)^123) is divided by 3? 17/3 has a remainder of 2, feeding in 2 into the problem how would I then find the remainder? Thanks again!
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Re: What is the remainder when ((55)^15!)^188 is divided by 17 [#permalink] New post 27 Apr 2013, 01:30
Richard0715 wrote:
Thanks Kris, I appreciated it! I am just trying to apply the concept to something else but how would I go about this problem? What is the remainder when ((17)^10)^123) is divided by 3? 17/3 has a remainder of 2, feeding in 2 into the problem how would I then find the remainder? Thanks again!





Kris01 wrote:
When a number a is divided by another number b, the quotient would be some number q and we would get a remainder r. Since we need to find the remainder, we can ignore the quotient and feed in the remainder. On feeding in the remainder and raising it to the power, we can find the total remainder.

Hope this helps! Let me know if you need any further notification.

Richard0715 wrote:
Why did you plug the remainder of 4 into the problem? I understand 55/17 gives remainder of 4 and I see where you went with the problem but I don't understand why you put the remainder back into the problem? Im lost!



I would always try to play with 1 wherever it is possible to reduce the time spent on calculations. In your question, 17 can be written as (18-1) where 18 is divisible by 3 and the rest about -1 is already explained by Kris01.

Hope this helps
Re: What is the remainder when ((55)^15!)^188 is divided by 17   [#permalink] 27 Apr 2013, 01:30
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