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# 2,600 has how many positive divisors?

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

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28 Oct 2010, 17:10
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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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28 Oct 2010, 17:21
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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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29 Oct 2010, 02:47
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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29 Oct 2010, 03:11
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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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29 Oct 2010, 03:27
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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29 Oct 2010, 23:57
Thanx Bunuel
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Re: 2,600 has how many positive divisors? [#permalink]

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20 Mar 2017, 01:27
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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03 Apr 2017, 00:16
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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05 Jan 2018, 11:24
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.

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

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05 Jan 2018, 11:56
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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

Cheers,
Brent
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Re: 2,600 has how many positive divisors?   [#permalink] 05 Jan 2018, 11:56
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