The remainder when 19[p]92[/p] is divided by 92 is

JEE Advanced 2015 Questions › Fundamentals questions › The remainder when 19[p]92[/p] is divided by 92 is

The remainder when 19⁹² is divided by 92 is

a) 41
b) 49
c)32
d)44

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Ricky ·2010-05-11 07:39:30

The idea is that , 19 ^{f ( p )} = 1 ( mod 92 ) , where f ( p ) represents Euler's totient function , i . e , it caculates the number of co - prime numbers less than or equal to p . Its value is , in this case ---

f ( P ) = 92 ( 1 - 1 / 2 ) ( 1 - 1 / 23 ) = 44

So , 19 ⁴⁴ = 1 ( mod 92 )

or , 1 9 ⁸⁸ = 1 ( mod 92 )

So , 19 ⁹² = 19 ⁴ ( mod 92 ) = 361 ² ( mod 92 ) = ( - 7 ) ² ( mod 92 ) = 49 ( mod 92 )

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Rajan kushwaha ·2021-09-05 13:18:00

Chinese Remainder theorem (along with other results).

First note 92= 4 Ã— 23 with gcd(4,23) =1.

Let us call N= 1992.

We will compute, N(mod 4) and N (mod 23) and then use CRT tocompute N (mod 92).

First, N (mod 4) = (19)92( ð‘šð‘œð‘‘ 4) = (âˆ’1)92(ð‘šð‘œð‘‘ 4) = 1and ð‘(ð‘šð‘œð‘‘ 23) = 194.[(19)22 (ð‘šð‘œð‘‘ 23)]2(ð‘šð‘œð‘‘ 23) = (âˆ’4)4(ð‘šð‘œð‘‘ 23) = (16)4(ð‘šð‘œð‘‘ 23) = (âˆ’7)2(ð‘šð‘œð‘‘ 23) = 49(ð‘šð‘œð‘‘ 23) = 3.

Note in the above we have used Fermatâ€™s Little Theorem.

Now, If you know CRT,

you candirectly say ð‘( ð‘šð‘œð‘‘ 92) = 49.

If not, you can compute it.

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The remainder when 19[p]92[/p] is divided by 92 is

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