Divisors of 55,125: Counting Principle

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    Counting Principle
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Discussion Overview

The discussion centers on determining the number of divisors of the number 55,125, specifically exploring the application of the counting principle in relation to its prime factorization.

Discussion Character

  • Mathematical reasoning, Homework-related

Main Points Raised

  • One participant states that 55,125 can be expressed as \( (3)^2 \cdot (5)^3 \cdot (7)^2 \) and questions how many divisors it has.
  • Another participant suggests that each divisor must be of the form \( 3^a \cdot 5^b \cdot 7^c \), where \( a, b, \) and \( c \) are integers greater than or equal to 0, and seeks to determine the total count of such combinations.
  • A different participant describes a lengthy calculation process involving specific values for \( a, b, \) and \( c \) and concludes with a count of 36 divisors, while expressing uncertainty about the efficiency of their method.
  • Another participant references the general formula for finding the number of divisors based on prime factorization, indicating that the exponents of the prime factors dictate the range for each exponent in the divisor's form.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the number of divisors, as there are differing methods and calculations presented, leading to uncertainty about the correct answer.

Contextual Notes

Some calculations and assumptions made by participants may depend on specific interpretations of the divisor counting method, and there may be unresolved steps in the calculations presented.

Who May Find This Useful

Individuals interested in number theory, particularly those studying divisor functions and prime factorization, may find this discussion relevant.

weiji
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How many divisors does 55,125 have? For example, 55,125 = (3)^2 . (5)^3 . (7)^2
 
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Welcome to PF!

Hi weiji! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
weiji said:
How many divisors does 55,125 have? For example, 55,125 = (3)^2 . (5)^3 . (7)^2

Well, each divisor has to be of the form 3a5b7c, wiht a b and c integers > 0 …

so how many is that? :smile:​
 
I did a very long calculation by assume a=1, b=1 ; a=1,c=1 ; b=1,c=1, from here, I know 1575x35 = 2625x21 = 3675x15. Then I calculate each possible answer, I got 36 divisors. But is there any faster way? I really have no idea. :(
 
If the prime factorization of [tex]n[/tex] is [tex]n=p_1^{e_1}p_2^{e_2}...p_n^{e_n}[/tex]. Now, in any divisor, each prime factor's exponent [tex]a[/tex] range from [tex]0\leq a \leq e_i[/tex].
 
Thanks for sharing. By the way, I'm new here and nice to meet you all.
 

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