Binomial Theorem
Grade 11
Question:
<p><span class="math-tex">\(25^{190}-19^{190}-8^{190}+2^{190}\)</span> is divisible by</p>
<p style="display:inline">Neither 14 nor 34</p>
<p style="display:inline">34 but not by 14</p>
<p style="display:inline">Both 14 and 34</p>
<p style="display:inline">14 but not by 34</p>
Step-by-Step Solution
Key Concept: Group the terms into pairs to utilize the divisibility property that $a^n - b^n$ is divisible by $(a-b)$ for all $n$ and by $(a+b)$ when $n$ is even.
<p>The given expression is divisible by 6 and 17.<br />
Also, <span class="math-tex">$25^{190}-8^{190}$</span> is not divisible by 7<br />
but <span class="math-tex">$19^{190}-2^{190}$</span> is divisible by 7,<br />
So, <span class="math-tex">$25^{190}-19^{190}-8^{190}+2^{190}$</span> is divisible by 34 but not by 14.</p>
Correct Answer: B