While answering Aptitude questions in a book I faced this question but not able to find the solution So I searched Google and got two answers but didn't get an idea how the answer came.
Question:
P is a natural number. 2P has 28 divisors and 3P has 30 divisors. How many divisors of 6P will be there?
Solution 1:
2P is having 28(4*7) divisors but 3P is not having a total divisors which is divisible by 7, so the first part of the number P will be 2^5.
Similarly, 3P is having 30 (3*10) divisors but 2P does not have a total divisors which is divisible by 3. So 2nd part of the number P will be 3^3. So, P = 2^5*3^3 and the solution is 35.
Solution 2:
2P has 28 divisors =4×7,
3P has 30 divisors
Hence P=2^5 3^3
6p =2^6 3^4
Hence 35 divisors
I have been trying to understand the steps but not able to get.
Best Answer
hint
Let the prime factorization of $P=2^{a} \cdot 3^{b} \cdot 5^{c} \cdot 7^{d} \ldots$. Then the number of divisors of $P$ is given by $$(a+1)(b+1)(c+1)\ldots.$$
If $2P$ has $28$ Divisors then $$(a+2)(b+1)(c+1)\ldots=28.$$
Likewise If $3P$ has $30$ Divisors then $$(a+1)(b+2)(c+1)\ldots=30.$$
I hope now you can understand the solutions you found. If not let me know I will elaborate.