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# the remainder when 5^99 is divided by 13

8 years ago

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)

8 years ago

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