UPSC CDS 1 2023 Math Paper

Concept:We use the negative remainder method: Express the base as a multiple of the divisor minus 1, then evaluate the power of −1.Explanation:First, find the remainder when 65 is divided by 11.$65 = 11 \times 6 - 1$, so the remainder is $-1$ (or $10$ if positive).Now, $65^{99}$ divided by 11 gives the same remainder as $(-1)^{99}$.Because $99$ is an odd number, $(-1)^{99} = -1$.A negative remainder is not acceptable; we convert it to a positive remainder by adding the divisor $11$.So, positive remainder $= 11 + (-1) = 10$.Alternatively, using the concept: For $(ax - 1)^n$ divided by $a$, remainder is $(-1)^n$. If $n$ is odd, remainder is $-1$, which corresponds to $a - 1$ as positive remainder. Here $a = 11$, so $a - 1 = 10$.Thus, the remainder when $65^{99}$ is divided by $11$ is $10$.Answer:$10$Option D is correct.