The question statement gives us a hint that p is a composite number. Therefore, we are working with a question testing us on composite number concepts.
From statement I alone, we know that 10p has 2 more distinct prime factors than 10 has. Let us look at 10.
10 = 2*5, which means 10 has 2 distinct prime factors. Therefore, we can conclude that 10p has 4 distinct prime factors. Of these 4, 2 and 5 are from the 10; therefore, the p should be contributing the remaining 2 distinct prime factors.
Knowing that p has 2 distinct prime factors, we can answer the question with a NO. Statement I alone is sufficient. The possible answers at this stage are A or D. Answer options B, C and E can be eliminated.
From statement II alone, we know that \((4p)^3\) has 3 more distinct prime factors than 18 has. Let us analyse 18.
18 = 2 * \(3^2\), which means 18 also has 2 distinct prime factors.
Now, let us look at \((4p)^3\). \((4p)^3\) is nothing but \(2^6\) * \(p^3\). Clearly, 2 is one of the distinct prime factors. But, as per the data given in statement II, \(2^6\) * \(p^3\) should have 5 distinct prime factors (3 more distinct prime factors than the ones in 18). This can only happen when p has 4 distinct prime factors.
Knowing that p has 4 distinct prime factors, we can answer the question with a YES. Statement II alone is sufficient. Answer option A can be eliminated.
The correct answer option is D.
Hope that helps!
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