 Posers and Puzzles

1. 16 Feb '07 18:45
Find the last digit in result of 2^1996.

Explanation how you did it would be nice. It's not that I don't know the answer, I would just like to see if there are other ways than I figured of solving this problem.
2. 16 Feb '07 19:00
Originally posted by kbaumen
Find the last digit in result of 2^1996.

Explanation how you did it would be nice. It's not that I don't know the answer, I would just like to see if there are other ways than I figured of solving this problem.
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The first five powers of 2 are: 2, 4, 8, 16, 32. The final digits will therefore cycle through 2, 4, 8, 6 (in that order). When the index is a multiple of 4 the final digit will be 6, hence 2^1996 has final digit of 6.
3. 16 Feb '07 19:23
Originally posted by kbaumen
Find the last digit in result of 2^1996.

Explanation how you did it would be nice. It's not that I don't know the answer, I would just like to see if there are other ways than I figured of solving this problem.
6?
4. 16 Feb '07 19:57
Correct.
5. 16 Feb '07 20:48
Originally posted by kbaumen
Correct.
It was an easy one.

The G

p.s. Check "Site Ideas" for my revolutionary pole. Please make sure you vote.
6. 16 Feb '07 20:50
Originally posted by GinoJ
p.s. Check "Site Ideas" for my revolutionary pole.
Ummm, no...I don't think I want to see your revolutionary "pole".
7. 16 Feb '07 23:30
Originally posted by PBE6
Ummm, no...I don't think I want to see your revolutionary "pole".
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8. 16 Feb '07 23:31
Originally posted by PBE6
Ummm, no...I don't think I want to see your revolutionary "pole".
😀
9. 18 Feb '07 16:08
I'll, do you 3 better..

The last 4 digits are 4336.
10. 18 Feb '07 17:08
Wow, I'm curious how did you figure that? Or is it just a random guess with being sure only about the last digit? Or you have a very advanced calculator?
11. 18 Feb '07 18:39
Modulo arithmetic. The last 4 digits go through a cycle of 500 "last 4 numbers", and thus all I have to do is find the 500th and move backwards.

To find that out, I determine that 2^500 modulo 625 is 1, and thus find the four digits that are divisible by 16 and 1 more than a multiple of 625. 625 modulo 16 is 1, so I'm looking for 625*15+1 or 9376.

Now if I divide this by 16, I get another number, 586. So the remainder of my answer when divided by 625 has to be 586 AND divisible by 16.

586 modulo 16 is 10, which means I want 6*625 + 586 which is 4336.

12. 18 Feb '07 19:01
Originally posted by geepamoogle
Modulo arithmetic. The last 4 digits go through a cycle of 500 "last 4 numbers", and thus all I have to do is find the 500th and move backwards.

To find that out, I determine that 2^500 modulo 625 is 1, and thus find the four digits that are divisible by 16 and 1 more than a multiple of 625. 625 modulo 16 is 1, so I'm looking for 625*15+1 or 9376.
...[text shortened]... dulo 16 is 10, which means I want 6*625 + 586 which is 4336.

13. 18 Feb '07 19:27