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2,600 has how many positive divisors?

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2,600 has how many positive divisors? [#permalink]

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New post 28 Oct 2010, 16:10
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2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48
[Reveal] Spoiler: OA

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Re: divsors [#permalink]

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New post 28 Oct 2010, 16:21
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By factorization, you can write 2600 as 2600=2^3*5^2*13^1. Number of factors = (3+1)(2+1)(1+1)=24

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Re: divsors [#permalink]

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New post 29 Oct 2010, 01:47
vgan4 wrote:
Number of factors = (3+1)(2+1)(1+1)=24


can' t understand how we came to this. What does this multiplication mean?

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Re: divsors [#permalink]

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New post 29 Oct 2010, 02:11
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ulm wrote:
vgan4 wrote:
Number of factors = (3+1)(2+1)(1+1)=24


can' t understand how we came to this. What does this multiplication mean?


Finding the Number of Factors of an Integer

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:
If p and q are prime numbers, how many divisors does the product \(p^3*q^6\) have?

According to above the number of distinct factors of \(2,600=2^3*5^2*13\) would be \((3+1)(2+1)(1+1)=24\).

For more on this check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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Re: divsors [#permalink]

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New post 29 Oct 2010, 02:27
Yes, thanks,
i forgot
"The number of factors of n will be expressed by the formula \((p+1)(q+1)(r+1)\)"

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Re: divsors [#permalink]

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New post 29 Oct 2010, 22:57
Thanx Bunuel :)

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Re: 2,600 has how many positive divisors? [#permalink]

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New post 20 Mar 2017, 00:27
shrive555 wrote:
2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48


2600 = 13 x 2^3 x 5^2

no of possible divisor = (1+1)(2+1)(3+1) = 4.3.2 = 24

Option D

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Re: 2,600 has how many positive divisors? [#permalink]

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New post 02 Apr 2017, 23:16
2600= 26*100= 2*13*2^2*5^2
=2^3*5^2*13

Total factors= (3+1)(2+1)(1+1)= 4*3*2=24
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Re: 2,600 has how many positive divisors? [#permalink]

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New post 05 Jan 2018, 10:24
shrive555 wrote:
2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48


To determine the number of positive divisors, we break 2,600 into primes, add 1 to the exponent of each unique prime, and then multiply those values together.

We see that 2,600 = 26 x 100 = 2 x 13 x 4 x 25 = 2^3 x 5^2 x 13^1.

Now we add 1 to each exponent and multiply those results:

(3 + 1)(2 + 1)(1 +1) = 24

Thus, 2,600 has 24 positive divisors.

Answer: D
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Re: 2,600 has how many positive divisors? [#permalink]

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New post 05 Jan 2018, 10:56
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shrive555 wrote:
2,600 has how many positive divisors?

A. 6
B. 12
C. 18
D. 24
E. 48


Experts, is there a fast way to find all the divisors? Thanks.[/quote]

If the prime factorization of N = (p^a)(q^b)(r^c) . . . (where p, q, r, etc are different prime numbers), then N has a total of (a+1)(b+1)(c+1)(etc) positive divisors.

Example: 14000 = (2^4)(5^3)(7^1)
So, the number of positive divisors of 14000 = (4+1)(3+1)(1+1) =(5)(4)(2) = 40
----------ONTO THE QUESTION-------------------------

2600 = (2)(2)(2)(5)(5)(13)
= (2^3)(5^2)(13^1)
So, the number of positive divisors of 2600 = (3+1)(2+1)(1+1) =(4)(3)(2) = 24

Answer: D

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Brent
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Re: 2,600 has how many positive divisors?   [#permalink] 05 Jan 2018, 10:56
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