Binomial Theorem
Grade 11
Question:
<p>The remainder when (2021)<sup>2022</sup> + (2022)<sup>2021</sup> is divided by 7 is</p>
<p style="display:inline">2</p>
<p style="display:inline">6</p>
<p style="display:inline">1</p>
<p style="display:inline">0</p>
Step-by-Step Solution
Key Concept: Express each base as a multiple of the divisor plus or minus a small remainder and use binomial expansion to simplify the expression into manageable terms.
<p>(2021)<sup>2022</sup> + (2022)<sup>2021</sup> [ <span class="math-tex">$\because$</span> 2023 = 7 <span class="math-tex">$\times$</span> 289]<br />
= (2023 - 2)<sup>2022</sup> + (2023 - 1)<sup>2021</sup><br />
= 7k<sub>1</sub> + 2<sup>2022</sup> + 7k<sub>2</sub> - 1 = 7(k<sub>1</sub> + k<sub>2</sub>) + 8<sup>674</sup> - 1<br />
= 7(k<sub>1</sub> + k<sub>2</sub>) + (7 - 1)<sup>674</sup> - 1 = 7(k<sub>1</sub> + k<sub>2</sub>) + 7k<sub>3</sub> + 1 - 1<br />
= k<sub>1</sub> + k<sub>2</sub> + k<sub>3</sub><br />
<span class="math-tex">$\therefore$</span> Given number is divisible by 7 hence remainder is zero.</p>
Correct Answer: D