- #1
weiji
- 7
- 0
How many divisors does 55,125 have? For example, 55,125 = (3)^2 . (5)^3 . (7)^2
I'm WeijiIn summary, the conversation discusses finding the number of divisors for the number 55,125, using the example of its prime factorization. The conversation also explores a method of calculating the number of divisors, with a final answer of 36 possible divisors.
- #2
tiny-tim
Science Advisor
Homework Helper
- 25,839
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Welcome to PF!
Hi weiji! Welcome to PF!
(try using the X2 tag just above the Reply box)
Well, each divisor has to be of the form 3a5b7c, wiht a b and c integers > 0 …
Hi weiji! Welcome to PF!
(try using the X2 tag just above the Reply box
)weiji said:
Well, each divisor has to be of the form 3a5b7c, wiht a b and c integers > 0 …
so how many is that?
- #3
weiji
- 7
- 0
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. :(
- #4
JSuarez
- 402
- 1
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].
- #5
weiji
- 7
- 0
Thanks for sharing. By the way, I'm new here and nice to meet you all.
Related to Divisors of 55,125: Counting Principle
1. What is the Counting Principle?
The Counting Principle is a mathematical concept that states that the total number of possible outcomes of a series of events is equal to the product of the number of outcomes for each event. In other words, if there are m possible outcomes for one event and n possible outcomes for another event, then the total number of outcomes for both events is m x n.
2. How does the Counting Principle apply to divisors of 55,125?
When determining the divisors of a number, we are essentially finding all the possible outcomes of dividing that number by smaller numbers. For example, the number 6 has 4 divisors: 1, 2, 3, and 6. This is because there are 4 possible outcomes of dividing 6 by smaller numbers (1, 2, 3, and 6). In the case of 55,125, the Counting Principle tells us that there are 3 x 5 x 5 x 5 = 375 possible outcomes of dividing 55,125 by smaller numbers, giving us a total of 375 divisors.
3. What is the largest divisor of 55,125?
The largest divisor of 55,125 is the number itself, 55,125. This is because any number divided by itself equals 1, and 1 is a divisor of all numbers.
4. How can I determine the number of divisors of a given number?
To determine the number of divisors of a given number, you can use the prime factorization method. First, find the prime factors of the number. Then, use the Counting Principle to determine the number of divisors. For example, the prime factorization of 55,125 is 3 x 5 x 5 x 5. Therefore, there are (1+1) x (1+1) x (1+1) x (1+1) = 16 divisors of 55,125.
5. Can the Counting Principle be applied to other mathematical concepts?
Yes, the Counting Principle can be applied to various mathematical concepts, such as permutations, combinations, and probability. It is a fundamental principle in combinatorics, which is the branch of mathematics that deals with counting and arranging objects. Understanding the Counting Principle can help in solving more complex mathematical problems involving combinations and arrangements.
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