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

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Answer

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Solution doesn't satisfie, can someone solve this correctly.

Answer

Coming to concepts of prime factorization

any number can be expressed as powers of products of prime numbers ,
also the thing to note is that, the number of divisors of a number thus can be obtained by this prime factorization, which is product of (powers+1)
let a=2^a*3^b....
so no of divisor =( a+1)(b+1)....

here let
p has r divisor
r=(a+1)(b+1)k
28=(a+2)(b+1)k. where k is contribution from other primes than 2 and 3

similarly,
30 =(a+1)(b+2)k
by trial and error
28=1*4*7
30=1*6*5
thus x=5
y =3
k=1
so 3p has (a+2)(b+2)k divisor
=35 divisor

Answer

Given:
- P is a natural number.
- 2P has 28 divisors.
- 3P has 30 divisors.

To find:
- The number of divisors of 6P.

Solution:
Let's first find the prime factorization of 2P and 3P. We can write:

2P = 2^a * q
3P = 3^b * r

where q and r are products of distinct primes other than 2 and 3.

Let's count the number of divisors of 2P and 3P using the formula:

Number of divisors = (a+1) * (b+1)

For 2P, we know that it has 28 divisors. So, we have:

(a+1) * (b+1) = 28

We also know that a is one less than the number of prime factors of 2P. Since 2P has only one prime factor of 2, we have:

a = 1

Substituting a = 1 in the equation (a+1) * (b+1) = 28, we get:

2 * (b+1) = 28

Simplifying, we get:

b+1 = 14
b = 13

So, we have:

2P = 2 * 13 * q

Similarly, for 3P, we have:

(a+1) * (b+1) = 30

Since 3P has only one prime factor of 3, we have:

b = 1

Substituting b = 1 in the equation (a+1) * (b+1) = 30, we get:

(a+1) * 2 = 30

Simplifying, we get:

a+1 = 15
a = 14

So, we have:

3P = 3^2 * r

Now, let's find the prime factorization of 6P. We know that:

6P = 2 * 3 * P

Substituting the prime factorizations of 2P and 3P, we get:

6P = 2 * 3 * 2 * 13 * q * 3^2 * r

Simplifying, we get:

6P = 2^2 * 3^3 * 13 * q * r

The number of divisors of 6P is given by:

Number of divisors = (2+1) * (3+1) * (1+1) * (1+1)

Simplifying, we get:

Number of divisors = 36

However, we need to exclude the divisors that contain factors of both 2P and 3P (i.e., factors of 6P that are also factors of 2P and 3P). These divisors are of the form:

2^i * 3^j * q * r

where i is between 0 and 1 (since we already have two factors of 2), and j is between 0 and 2 (since we already have two factors of 3). There are 2 * 3 = 6 such divisors.

Therefore, the number of divisors of 6P that are not divisors of 2P or

Answer

How you write 2p and 3p....will you please give any explanation....

Answer

Answer is A

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