hey ..............sorry................but...........as it is a binomial theorem question.................. but i m doing it by different method...........

 

thanku...............

look.........

2=2  (mod 7)                 (means when 2 / 7 leaves remainder 2)

2⁵ = 32 (mod 7)

2⁵ = 4  (  mod 7)              ( 32 divided  by 7 gives 4)

32³² = 4 ³²  (mod 7)          ( raising power of 32 on both sides)

32³² =  2⁶⁴  (mod 7)  ....................  (  1)

 

now lets find  remainder obtained by 7 of 2⁶⁴

2 = 2 ( mod 7)

2² =4  (mod 7)

2⁴  = 2 ( mod 7)            (as 16 divided by 7 gives remainder 2)

2⁸ =  2²  =4  (mod 7)

2¹⁶ = 2  ( mod 7)   ( as 16 divided by 7 gives 2 rem)

2³² = 4 ( mod 7)................................. ( 2  )

 

2 ⁶⁴ = 2 ( mod 7)   ( as 16 gives 2 rem)..............( 3)

 

putting (3)   in    (1)............... we get ............

 

32³²= 2   ( mod 7 )

 

₃₂32³² = 2³² 

 now putting 2 in 1...............

we get

₃₂32³²= 4  (  mod 7)..........................

 

 

ya.. ya i know it its sometime confusing...................but............i have tried my best to explain it hope u understands...........