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If r, s, and t are all positive integers, what is the

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If r, s, and t are all positive integers, what is the [#permalink] New post 29 Jan 2007, 16:37
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If r, s, and t are all positive integers, what is the remainder of (2^p)/10 , if p=rst?

(1) s is even

(2) p = 4t

please provide explanations for answers.
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Re: DS-Remainder [#permalink] New post 29 Jan 2007, 17:36
buckkitty wrote:
If r, s, and t are all positive integers, what is the remainder of (2^p)/10 , if p=rst?

(1) s is even

(2) p = 4t

please provide explanations for answers.



B for me.

Stmt 1) Dont know about r and t so remainder can be anything

Stmt 2) p = 4t

try putting some values for t 1,2 ,3 in every case the remainder is 6.

So sufficient.
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 [#permalink] New post 29 Jan 2007, 18:03
My ans is B

the last digit of 2^p should be 2, 4, 8,or 6, so the remainder should be one of these.

S1: p is even
so the remainder should be 4 or 6 --- insuff

S2: p should be a multiple of 4
The last digit of 2^4, 2^8... should be 6
so the remainder is 6 --- suff
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 [#permalink] New post 30 Jan 2007, 09:27
Good answers. OA is B

Is the reason why we know that p=4t is enough because there are only 4 possible remainders when 2^p is divided by 10 (2, 4, 8, 6)? Therefore 4 times a number will always end up in the 4th spot? In other words, if (B) said p=3t then it would be insufficient because every multiple of 3t would result in a different remainder (until you cycled through all 4 possibilities).

Plugging in is good here too, because it only takes a couple options to realize the remainder is always the same.
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 [#permalink] New post 30 Jan 2007, 11:13
Quote:
the last digit of 2^p should be 2, 4, 8,or 6, so the remainder should be one of these.
are their any other remainder generalizations such as this? I would've had to of done n amt. of probs like this to finally realize that.
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 [#permalink] New post 30 Jan 2007, 11:25
ggarr wrote:
Quote:
the last digit of 2^p should be 2, 4, 8,or 6, so the remainder should be one of these.
are their any other remainder generalizations such as this? I would've had to of done n amt. of probs like this to finally realize that.


Not necessarily for remainders, but it is good to know that a pattern occurs for every units digit that is raised to the nth power.

There is a pattern created. It is good to learn some and know that there is a pattern for all.
pattern of units digit when a number is raised to ^n
2^n 2,4,8,6
3^n 3,9,7,1
4^n 4,6,4,6
5^n 5,5,5,5
...and so on....

knowing the patterns exist helps with remainder questions
  [#permalink] 30 Jan 2007, 11:25
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