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  • If 2298 is divided by 63 then which of the following is true?Option: 1 Remainder is 1Option:

If 2298 is divided by 63 then which of the following is true?

Option: 1

Remainder is 1


Option: 2

Integer just less than the number \frac{2^{298}}{63} is odd


Option: 3

Remainder is even


Option: 4

None of the above is true


Answers (1)

best_answer

Finding Remainder Using Binomial Theorem

If the number is given in the form of ‘an’ and which is divided by ‘b’. To find the remainder, adjust the power of ‘a’ to am such that it is very close to ‘b’ with a difference of 1 (i.e. b + 1 or b - 1).  

 

Now,

\begin{aligned} 2^{298} &=2^{4} \cdot\left(2^{6}\right)^{49} \\ &=4 \times(64)^{49}=4 \times(63+1)^{49} \end{aligned}

Using binomial expansion

2^{298}=4 \times\left[^{49} C_0\;(63)^{49}+\;^{49} C_{1}(63)^{48}\cdots\cdots\;+^{49} C_{49}(1)^{49}\right]

\begin{aligned}{2^{298}={4 \times[63 k+1]}} \\ {=252 k+4}\end{array}

\begin{aligned} \frac{2^{298}}{63} &=\frac{252 k}{63}+\frac{4}{63} \\ &=4 k+\frac{4}{63} \end{aligned}

hence, the quotient part is even and the remainder is 4 (even). 

option C is correct

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Pankaj

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