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Evaluate this remainder (Posted on 2008-08-13) Difficulty: 2 of 5
What is the remainder when you divide 299 by 99?

  Submitted by pcbouhid    
Rating: 4.0000 (1 votes)
Solution: (Hide)
Finding the remainders when successive powers of 2 are divided by 99:
    power of 2   remainder
   ------------ -----------
         1            2
         2            4
         3            8
         4           16
         5           32
         6           64
         7           29
         8           58
         9           17
        10           34
        11           68
        12           37
        13           74
        14           49
        15           98
        16           97
        17           95
        18           91
       ...          ...
The remainder when 2^15 is divided by 99 is 98, so:

2^15 = 99n + 98 or
2^15 = 99(n+1) - 1

thus the remainder 98 can be considered as a remainder of -1.

When I multiply 2^15 times 2^15, the remainder will be (-1) times (-1), which is 1.

So I know that:

(2^15)(2^15) = 2^30 has a remainder 1 when divided by 99.

We have:

2^99 = (2^30)(2^30)(2^30)(2^9)

And, since the remainder when 2^30 is divided by 99 is 1, the remainder when 2^99 is divided by 99 is the same as the remainder when 2^9 is divided by 99 - and we found in the table above that this remainder is 17.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Solutionusing CRT and Eulers-Fermat TheoremsPraneeth2008-08-14 01:54:04
Solutionre: Rereadingbrianjn2008-08-13 21:55:25
SolutionAlternative MethodologyK Sengupta2008-08-13 11:38:11
SolutionSolutionK Sengupta2008-08-13 11:36:25
Solutionre: Rereading (solution; spoiler)Charlie2008-08-13 11:14:51
Rereadingbrianjn2008-08-13 10:42:23
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