Binomial Theorem
Grade None
Question:
<p>The remainder when (11)<sup>1011</sup> + (1011)<sup>11</sup> is divided by 9 is</p>
<p style="display:inline">4</p>
<p style="display:inline">8</p>
<p style="display:inline">1</p>
<p style="display:inline">6</p>
Step-by-Step Solution
Key Concept: Reduce the bases using modular arithmetic and express the powers in terms of multiples of the divisor plus or minus one to simplify the remainder calculation.
<p>Given that (11)<sup>1011</sup> + (1011)<sup>11</sup><br />
= (9 + 2)<sup>1011</sup> + (1008 + 3)<sup>11</sup><br />
<span class="math-tex">\(\Rightarrow 9\lambda\)</span> + 2<sup>1011</sup> + <span class="math-tex">\(9\mu\)</span> + 3<sup>11</sup> <span class="math-tex">\(\Rightarrow\)</span> (2<sup>3</sup>)<sup>337</sup> + 3<sup>2</sup> <span class="math-tex">\(\times\)</span> 3<sup>9</sup> = 8<sup>337</sup><br />
<span class="math-tex">\(\Rightarrow\)</span> (9 - 1)<sup>337</sup><sup> </sup>= 9K - 1<sup>337</sup> <span class="math-tex">\(\therefore\)</span> Remainder = 9 - 1 = 8</p>
Correct Answer: B