Latest Campus Aptitude Question SOLUTION: Find the remainder when 83^261 is divided by 17

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Find the remainder when 83^261 is divided by 17

Asked In Campus Dada Khalandar (10 years ago)

Unsolved Read Solution (5)

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Saraswathy  (10 years ago)

83^261 mod 17

(17*5-2)^261 % 17
(-2)^261 % 17 [Since 17*5%17 = 0]
Now 2 nd 17 are co-primes... So we can use Euler's theorem...
17 => 17^1 [a^p*b^q*..]
E[17]=16 [ Since E(N)=N*(1-1/a)*(1-1/b)...]

Therefore we can write
[(-2^16)^16 * (-2)^5] % 17
1^16 * (-32) % 17 [Since Rem[M^E(N)/N] = 1]
-32 % 17
(-32+17) % 17 [Since rem can't be negative]
-15 % 17
(-15+17) % 17
2 % 17

Hence reminder is 2...

Ans : 2

Is this solution Helpfull? Yes (15) | No (1)  

Saraswathy  (10 years ago)

83^261 mod 17

(17*5-2)^261 % 17
(-2)^261 % 17 [Since 17*5%17 = 0]
Now 2 nd 17 are co-primes... So we can use Euler's theorem...
17 => 17^1 [a^p*b^q*..]
E[17]=16 [ Since E(N)=N*(1-1/a)*(1-1/b)...]

Therefore we can write
[(-2)^16 * (-2)^16 * (-2)^5] % 17
1 * 1 * (-32) % 17 [Since Rem[M^E(N)/N] = 1]
-32 % 17
(-32+17) % 17 [Since rem can't be negative]
-15 % 17
(-15+17) % 17
2 % 17

Hence reminder is 2...

Ans : 2

Is this solution Helpfull? Yes (1) | No (1)  

Uttam Kumar  (10 years ago)

83^261 mod 17
solution-
83^4 contain unit digit is 1;
so 83^(261/4) mod 17;
now 83^(1) Write only reminder on power;
(261/4=65 is quotient and 1 is reminder)
now only 83/17=4 quotient and 15 is reminder
ans = 15 ;

Is this solution Helpfull? Yes | No (2)  

Rahul Das  (10 years ago)

83^261%17
(17*5-2)^261 % 17
(-2)^261 % 17 [Since 17*5%17 = 0]
now 261%4=1 we can write 261 as 4n+1
so that
(-2)^1/17
=-2+17=15
so remainder will be 15

Is this solution Helpfull? Yes | No (2)  

vijeta kumari  (9 years ago)

if 261 is divided by 4 then remainder will be 1
so, 83^261 became 83^1
now 83^1 is divided by 17
then remainder will be 15

Is this solution Helpfull? Yes | No (2)  

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