Question

Please read the following sentences carefully:
I — 103 and 7 are the only prime factors of \(1000027\).
II — \(\sqrt[6]{6!} \ge \sqrt[7]{7!}\).
III — If I travel one half of my journey at an average speed of \(x\) km/h, it will be impossible for me to attain an average speed of \(2x\) km/h for the entire journey.

• A. All the statements are correct
• B. Only Statement II is correct
• C. Only Statement III is correct
• D. Both statements I and II are correct
• E. Both statements I and III are correct

Hint:

For comparing \(n\)-th roots, raise both sides to the LCM of exponents to avoid roots. In average-speed problems over equal distances, the harmonic mean logic shows that hitting a higher target average after a slow first half can be impossible.

Answer 1

The Correct Option is C

Step 1 (Statement I): \(1000027=10^6+27\). Check factors:
\[ 1000027=7\times 142861, \; 142861=103\times 1387, \; 1387=19\times 73. \]
Hence \(1000027=7\cdot 103\cdot 19\cdot 73\). So I is false.

Step 2 (Statement II): Compare \(\sqrt[6]{6!}\) and \(\sqrt[7]{7!}\). The claim \(\sqrt[6]{6!}\ge \sqrt[7]{7!}\) is equivalent to
\[ (6!)^7 \ge (7!)^6=(7\cdot 6!)^6=7^6(6!)^6 \;\Longleftrightarrow\; 6! \ge 7^6, \]
but \(6!=720 < 7^6=117{,}649\). Hence \(\sqrt[6]{6!}<\sqrt[7]{7!}\); II is false.

Step 3 (Statement III): Let total distance be \(D\). If the first half \(D/2\) is done at speed \(x\), time used is \(\tfrac{D}{2x}\). To average \(2x\) over the whole trip, total time must be \(\tfrac{D}{2x}\). This leaves zero time for the remaining half—impossible unless the remaining speed is infinite. So III is true.

Conclusion: Only Statement III holds.