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CAT 2003 Question(Finding the Remainder)

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Q.)What is the remainder when ${ 4 }^{ 96 }$ is divided by 6?
A)0
B)2
C)3
D)4

Solution : ${ 4 }^{ 96 }$ can also be written as

${ 4 }^{ 96 }=(6-2)^{ 96 }$

$Rem(\frac { { 4 }^{ 96 } }{ 6 } )=Rem(\frac { (6-2)^{ 96 } }{ 6 } )$

=$Rem(\frac { (2)^{ 96 } }{ 6 } )$

Now, If we divide Numerator and Denominator by 2, then find out remainder , then in the remainder we will have to multiply by 2.

So, dividing Numerator and Denominator by 2 , We get

$Rem(\frac { (2)^{ 96 } }{ 6 } ) = Rem(\frac { (2)^{ 95 } }{ 3 } )$

= $Rem(\frac { (3-1)^{ 95 } }{ 3 } )$

= $Rem(\frac { (-1)^{ 95 } }{ 3 } )$

= -1 which means the remainder will be 3-1 = 2

So, the actual remainder = 2X2 = 4(Because we had divided by 2 in Numerator and Denominator)

Hence, OPTION(D)