Q.4) Binomial Theorem

Q : Fractional part of 2^(78 / 31) is ______?
SOLUTION :
Since 2^78 = 2^75 * 2^3


=> 2^78 = (2^5)^15 * 2^3

=> 2^78 = (32)^15 * 2^3

=> 2^78 = (31 + 1)^15 * 2^3


Now, expanding (31 + 1)^15 using Binomial Theorem,

(31 + 1)^15 = C0 * (31^15) + C1 * (31^14) *1 + terms of descending powers of 31. . .+ Cn * 31^0 *1^15 . . {where, C0 , C1, . . Cn are binomial coefficients. }

=> (31 + 1)^15 = (sum of the terms divisible by 31) + 1


(2^78) / 31 = 2^3 * [ (sum of the terms divisible by 31) + 1] / 31

=> 2^3 * (sum of the terms divisible by 31) / 31 + (2^3) / 31

=> 8k + 8 / 31- - - - { where k is any integer}


so, the fractional part = 8 / 31 => (Ans)