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Find the remainder of the division 2^87/9 a. 8 b. 1 c. -1 d.

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Find the remainder of the division 2^87/9 a. 8 b. 1 c. -1 d. [#permalink] New post 07 Jan 2004, 13:16
Find the remainder of the division 2^87/9

a. 8
b. 1
c. -1
d. None of these.

***I got answer choice (c) -1.
The official answer was (a). Whose correct? any idea
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 [#permalink] New post 07 Jan 2004, 13:33
A is the answer. Look at the pattern:
2^1 = 2 2^5 = 32
2^2 = 4 2^6 = 64
2^3 = 8 2^7 = 128
2^4 = 16 2^8 = 256

Look at the pattern of the units digits. It repeats after every 4. Therefore, the exponent, 87/4 gives 21 remainder 3. A remainder of 3 means that the unit digit will be an 8 at the end. When you divide that unit's digit by 9, you will get a remainder of 8. Hope it helps
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Last edited by Paul on 07 Jan 2004, 13:45, edited 1 time in total.
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 [#permalink] New post 07 Jan 2004, 13:38
Yep, 8 it is.

2^1 / 9 is 0 remainder 2
2^2 / 9 is 0 remainder 4
2^3 / 9 is 0 remainder 8 .....

for 2^1 to 2^6, the respective remainders are 2,4,8,7,5,1
after the sixth exponent, you find that you have yourself a cycle.

So, 87/6 is 14 remainder 3. Excluding 14 complete cycles, the third position is 8 which is the remainder.

Don't know if I've made sense but...
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 [#permalink] New post 07 Jan 2004, 13:43
ndidi204 wrote:
Yep, 8 it is.

2^1 / 9 is 0 remainder 2
2^2 / 9 is 0 remainder 4
2^3 / 9 is 0 remainder 8 .....

for 2^1 to 2^6, the respective remainders are 2,4,8,7,5,1
after the sixth exponent, you find that you have yourself a cycle.

So, 87/6 is 14 remainder 3. Excluding 14 complete cycles, the third position is 8 which is the remainder.

Don't know if I've made sense but...


perfect...I did it up till getting the 2 to 1 cycle, but couldn't figured out what to do next.. yep 8 it is.
good work..!

hey, sunniboy, how did you get -1 as a remainder.. :idea: :?: :idea:
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 [#permalink] New post 07 Jan 2004, 13:47
k, this was my reasoning:

Using the formula for remainder theorem: I concluded that

9 = 2^3 + 1

2^87 = (2^3)^29

Therefore, let 2^3 = x

So, 9 = x +1; f(-1)

(-1)^29 = -1
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 [#permalink] New post 07 Jan 2004, 13:51
sunniboy007 wrote:
k, this was my reasoning:

Using the formula for remainder theorem: I concluded that

9 = 2^3 + 1

2^87 = (2^3)^29

Therefore, let 2^3 = x

So, 9 = x +1; f(-1)

(-1)^29 = -1


I don't see any point for considering it as a negative function. Can very well be +ve.
  [#permalink] 07 Jan 2004, 13:51
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Find the remainder of the division 2^87/9 a. 8 b. 1 c. -1 d.

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