Problems in finding divisors using permutation,combination

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In summary, to find the total number of divisors of a given number using permutations and combinations, we add 1 to the powers of each prime factor and then multiply them together. This is because for each prime factor, we have multiple options for its power in a divisor. By using this method, we can easily determine all possible divisors and find the total number of divisors.
  • #1
Shafia Zahin
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I am facing some problems when I am told to find the total number of divisors of a certain number using permutations and combinations.Here is an example:
What is the total number of divisors of the number 120?
Ans:
120=2^3*3^1*5^1
The total number of divisors:(3+1)(1+1)(1+1)=16

I just don't get it why do we add 1 with the powers of the divisors and then multiply them?And by doing this how can we find the total number of divisors?I have just become fully confused and unable to understand the mechanism.Can anyone please help me to understand this?
With regards,
Shafia.
 
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  • #2
Actually its like this
In any divisor of the number 120, 2 can occur in four ways namely: 2^0, 2^1, 2^2, 2^3
Similarly 3 can occur as : 3^0, 3^1
5 can occur as : 5^0, 5^1
Hence (3+1)*(1+1)*(1+1)= 16
 
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1. What is the difference between permutation and combination?

Permutation refers to the arrangement of a set of objects in a specific order, while combination refers to the selection of a group of objects from a larger set regardless of their order.

2. Why is finding divisors using permutation and combination a problem?

Finding divisors using permutation and combination is a problem because it involves a large number of calculations and can be time-consuming. It also requires a deep understanding of mathematical concepts such as factorization and combinatorics.

3. Can computer algorithms be used to solve this problem?

Yes, computer algorithms can be used to solve this problem. There are various algorithms such as the Sieve of Eratosthenes and the Pollard's rho algorithm that can efficiently find divisors using permutation and combination.

4. Are there any real-world applications for this problem?

Yes, there are real-world applications for finding divisors using permutation and combination. It is used in cryptography to generate large prime numbers, in computer science to optimize code, and in statistics to calculate probabilities and combinations in data analysis.

5. How can this problem be solved efficiently?

This problem can be solved efficiently by using mathematical formulas and algorithms specifically designed for finding divisors. It is also important to have a good understanding of number theory and combinatorial mathematics to find the most efficient solution.

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