Basic Mathematics & Logarithm
Logarithmic Inequalities
Grade 11
Question:
<p>The lengths of the sides of a triangle are \(\log_{10} 12\), \(\log_{10} 75\) and \(\log_{10} n\), where \(n \in \mathbb{N}\). If a and b are the least and greatest values of n respectively, the value of \(b - 7a\) is divisible by</p>
<p>(a) 221</p>
<p>(b) 222</p>
<p>(c) 223</p>
<p>(d) 224</p>
Step-by-Step Solution
Key Concept: Apply triangle inequality to logarithmic side lengths to find the range of valid natural numbers.
<p><strong>Solution:</strong> In a triangle, the triangle inequality must hold:</p><p>$\log_{10} 12 + \log_{10} 75 > \log_{10} n$</p><p>This gives: $\log_{10}(12 \times 75) > \log_{10} n$</p><p>Therefore: $n < 900$ ... (i)</p><p>Similarly, $\log_{10} 75 - \log_{10} 12 < \log_{10} n$</p><p>This gives: $\log_{10}\frac{75}{12} < \log_{10} n$</p><p>Therefore: $n > \frac{25}{4}$ ... (ii)</p><p>Since $n \in \mathbb{N}$, we have $a = 7$ and $b = 899$</p><p>Thus $b - 7a = 899 - 49 = 850$, which is divisible by 223.</p><p>∴ Answer is (c).</p>
Correct Answer: c