# P, q, and r are prime #'s and a, b, and c are positve intgers, how many divisors

• mr_coffee
In summary, the conversation involved discussing the number of possible divisors for a given equation involving prime numbers and positive integers. The correct answer is (a+1)(b+1)(c+1), but there was a typo in the original response. The correct solution involves considering all possible combinations of the listed numbers, which results in (a+1)(b+1)(c+1) divisors.
mr_coffee
Hello everyone.

I think i got this right but i want to make sure...
If p, q and r are prime numbers and a, b, and c are positive integers, how many possible divisors does p^a*p^b*r^c have?

I said...
There are a+1 divisors: 1, p, p^2...,p^a
A divisor is a product of anyone of the a+1 numbers listed above times anyone of the b+1 numbers 1, q, q^2...q^b. We also have r+1...1, r, r^2...,r^c so by the muliplication rule, there are (a+1)(b+1)(r+1) divisors in all.

Yes, that looks good, except for one part. The answer should be (a+1)(b+1)(c+1). You wrote (r+1) instead of (c+1), but I think this was just a typo.

ahh yes! thanks for picking that up, it was a typo.
Thanks for the responce.

## 1. What are prime numbers and how are they different from other numbers?

Prime numbers are numbers that are only divisible by 1 and themselves. They are different from other numbers because they have only two factors, while other numbers can have multiple factors.

## 2. How do you determine if a number is prime or not?

A number is determined to be prime if it has exactly two factors, 1 and itself. This means that it cannot be evenly divided by any other number.

## 3. How do you find the divisors of a given number?

To find the divisors of a given number, you need to factorize the number and then list all the possible combinations of its factors. For example, the divisors of 12 are 1, 2, 3, 4, 6, and 12.

## 4. How do P, q, and r being prime numbers and a, b, and c being positive integers affect the number of divisors?

If P, q, and r are prime numbers, then the number will have a total of (a+1)(b+1)(c+1) divisors, where a, b, and c are the powers of P, q, and r, respectively. This is because each prime factor can be raised to different powers to form different combinations of divisors.

## 5. How many divisors does a number with three prime factors and powers of 2, 3, and 5 have?

If a number has three prime factors and the powers are 2, 3, and 5, then the number will have (2+1)(3+1)(5+1) = 72 divisors. This is because the number can be written as 2^a x 3^b x 5^c, where a, b, and c can take on values of 0, 1, or 2, resulting in 3 x 4 x 6 = 72 possible combinations of divisors.

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