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InterviewSolution
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If the fractional part of the number `(2^(403))/(15)` is `(k)/(15)`, then k is equal toA. 14B. 6C. 4D. 8 |
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Answer» Consider , `2^(403)=2^(400+3)=8*2^(400)=8*2^(400)=8*(2^(4))^(100) =8*(16)^(100)=8(1+15)^(100) =8(1+""^(100)C_(1)(15)+""^(100)C_(2)(15)^(2)+....+""^(100)C_(100)(15)^(100))` [By binomial theorem , `(1+x)^(n) =""^(n)C_(0)+""^(n)C_(1)x+""^(n)C_(2)x^(2)+...""^(n)C_(n)x^(n) , n in N ]` `=8+8(""^(100)C_(1)(15)+""^(100)C_(2)(15)^(2)+...+""^(100)C_(100)(15)^(100))` `=8+8xx 15 lambda ` where `lambda=""^(100)C_(1)+....+""^(100)C_(100)(15)^(99)in N ` `:. (2^(403))/(15)=(8+8xx 15 lamda)/(15)=8lambda+(8)/(15)` `implies {(2^(403))/(15)}=(8)/(15)` (where `{*}` is the fractional part function ) `:. k=8` Alternate Method `2^(403)=8.2^(400)=8(16)^(100)`. Note that, when 16 is divided by 15, gives remainder 1. `therefore` When `(16)^(100)` is divided by 15, gives remainder `1^(100)=1` and when `8(16)^(100)` is divided by 15, gives remainder 8. `therefore {(2^(403))/(15)}=(8)/(15)`. (where `{.}` is the fractional part function) `rArr k=8 |
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