Obtaining the number of factors from prime factorization

In summary, the number of distinct factors of a number is determined by the sum of the kth powers of the divisors, where k is the number of prime factors. In the case of 2520, there are 48 distinct factors. This can be calculated using the divisor function, which takes into account the powers of the prime factors.
  • #1
danne89
180
0
Hi! How do I determine the number of distinct factors of a number, say, 2520?
2520 = 2*2*2*3*3*5*7
So we've 8 different primes. The number of combinations of those is, according to me:
C(8,1)+C(8,2)+...+C(8,8)=155 (I think, calculated it by hand; but it isn't important)
Obviously those aren't distinct. (Pick the fist 2 and the second 2 = 4, but pick the second 2 and the third 2 also = 4.) :yuck:
 
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  • #2
I don't understand what you are doing. 2520= 2*2*2*3*3*5*7 has 4 distinct prime factors, not "8 different primes". And I don't see what being "prime factors" has to do with number of combinations. Are you asking "of those 8 numbers (not all distinct) how many combinations can I make"? Wouldn't that be the same as asking "of the 8 letters "aaabbcd", how many different combinations can I make?" There are only 3 different one letter combinations: "a", "b", and "c", not C(8,1)= 8.
 
  • #3
Isn't the combinations of the prime factors = all factors?
I mean, i pick 2*2 (a product of the primes in the positions specified by the combination) or 12
i pick 2*2 or 23
i pick 2*3 or 24

and so on.
Do you see?
 
  • #4
danne89 said:
Hi! How do I determine the number of distinct factors of a number, say, 2520?
2520 = 2*2*2*3*3*5*7
So we've 8 different primes. The number of combinations of those is, according to me:
C(8,1)+C(8,2)+...+C(8,8)=155 (I think, calculated it by hand; but it isn't important)
Obviously those aren't distinct. (Pick the fist 2 and the second 2 = 4, but pick the second 2 and the third 2 also = 4.) :yuck:
This is a special case of the divisor functions studied in number theory where you sum the kth power of the divisors.
In this case the 0th power.
say your number factors as
p1^n1*p2^n2*p3^n3*...*pk^nk*...
a general factor (including improper ones) is
p1^m1*p2^m2*p3^m3*...*pk^mk*...
where pk is a prime and 0<=mk<=nk
Thus the number of such factors is
(1+n1)(1+n2)(1+n3)...(1+nk)...
In particular 1 has one factor and and p^n (p prime) has n+1
2520 has 4*3*2*2=48
see this link for more info
http://mathworld.wolfram.com/DivisorFunction.html
 

1. How do you obtain the number of factors from prime factorization?

To obtain the number of factors from prime factorization, you first need to list out all the prime factors of the given number. Then, you need to find the powers of each factor by dividing the given number by that factor repeatedly until you get 1. Finally, add 1 to each power and multiply all the results together to get the total number of factors.

2. Why is it important to know the number of factors from prime factorization?

Knowing the number of factors from prime factorization is important in mathematics because it helps in determining the divisibility of a number, finding the greatest common divisor and lowest common multiple of two numbers, and solving problems involving factors and multiples.

3. Can you explain the concept of prime factorization?

Prime factorization is the process of breaking down a composite number into its prime factors. A prime factor is a number that can only be divided by 1 and itself without leaving a remainder. By listing out all the prime factors of a number and their powers, we can express the number as a product of primes, also known as its prime factorization.

4. Is there a specific method for obtaining the number of factors from prime factorization?

Yes, there is a specific method for obtaining the number of factors from prime factorization, known as the fundamental theorem of arithmetic. This theorem states that every composite number can be expressed as a unique product of primes, and the total number of factors is equal to the product of the powers of each prime factor plus one.

5. Can prime factorization be used for large numbers?

Yes, prime factorization can be used for large numbers, but it can be a time-consuming process. However, there are algorithms and computer programs that can quickly find the prime factorization of extremely large numbers. Prime factorization is also useful in cryptography for secure encryption and decryption of messages.

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