P is a natural number. 2P has 28 divisors and 3P has 30 divisors. How many divisors of 6P will be there?a)35b)40c)45d)48Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Why is it taken as 2^6 and 3^3?

Answer

I didn't get the solution, as to how it is taken as 2^6 * 3^3 ?

Answer

ElBorate this answer please ..why it is taken as p=2^5

Answer

Understanding the Problem
We have two expressions involving the natural number $ P $:
- $ 2P $ has 28 divisors.
- $ 3P $ has 30 divisors.
We need to find the number of divisors of $ 6P $.

Divisor Function and Prime Factorization
The number of divisors function $ d(n) $ for a number $ n = p_1^{e_1} p_2^{e_2} \ldots p_k^{e_k} $ is given by:
$$ d(n) = (e_1 + 1)(e_2 + 1) \ldots (e_k + 1) $$
Assuming the prime factorization of $ P $ is:
$$ P = 2^{a} \cdot 3^{b} \cdot p_1^{c_1} \cdots p_k^{c_k} $$

Analyzing $ 2P $ and $ 3P $
For $ 2P $:
- $ 2P = 2^{a+1} \cdot 3^b \cdot p_1^{c_1} \cdots p_k^{c_k} $
- Divisors: $ (a+2)(b+1)(c_1+1)\cdots(c_k+1) = 28 $
For $ 3P $:
- $ 3P = 2^{a} \cdot 3^{b+1} \cdot p_1^{c_1} \cdots p_k^{c_k} $
- Divisors: $ (a+1)(b+2)(c_1+1)\cdots(c_k+1) = 30 $

Setting Up Equations
We can set up the following equations based on the divisor counts:
1. $ (a+2)(b+1)(c_1+1)\cdots = 28 $
2. $ (a+1)(b+2)(c_1+1)\cdots = 30 $

Finding $ 6P $
Now, for $ 6P $:
- $ 6P = 2^{a+1} \cdot 3^{b+1} \cdot p_1^{c_1} \cdots p_k^{c_k} $
The number of divisors will be:
$$ d(6P) = (a+2)(b+2)(c_1+1)\cdots(c_k+1) $$

Calculating $ d(6P) $
Using the equations derived earlier, we find:
Using the values $ (a+2)(b+1) = 28 $ and $ (a+1)(b+2) = 30 $, we can substitute and solve to find $ d(6P) $.
Through calculations, we find:
$$ d(6P) = 35 $$
Thus, the number of divisors of $ 6P $ is $ \textbf{35} $.

Conclusion
The correct answer is option 'A', which is \(

Answer

35

Answer

A

Answer

A) 35

Top Courses for Quant

Suggested Free Tests

Related Content

Quant Courses

Related Exams

Related Question Answers

Related Searches

Company