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the remainder when 5^99 is divided by 13
8 is the remainder.
using the concept of mod...
25 is conguent to -1 mod 13(means when -1 is subtracted from 25 it is divisible by 13)
5^2 is congruent to -1 mod 13
5^98 is congruent to -1 mod 13
hence, 5^98=13m-1
multipling by 5.....5^99=13.5.m - 5
so... 5^99=13n -13 + 8
5^99=13(n-1) +8
5^99+13m + 8
hence 8 is the remainder (ANS)
Dear Kaushik,
You can approach the problem in this way,
(5^2+1) is divisible by 13
and (5^2+1) divides [(5^2)^49+1^49] since (a+b)divides (a^n+b^n) for odd n
therefore 13 divides 5^98+1
therfore 13 divides [5(5^98+1)]
therefore 13 divides 5^99+5
Hencethe remainder is 5
hope I have come to your help
With regards ADHIRAJ
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