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# For how many positive integer x is 130/x an integer?

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Math Expert
Joined: 02 Sep 2009
Posts: 51262
For how many positive integer x is 130/x an integer?  [#permalink]

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12 Oct 2017, 22:55
00:00

Difficulty:

45% (medium)

Question Stats:

61% (00:53) correct 39% (00:57) wrong based on 76 sessions

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For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

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For how many positive integer x is 130/x an integer?  [#permalink]

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13 Oct 2017, 04:28
Bunuel wrote:
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

This question is a version of "find the number of factors." An integer divided by any of its factors equals an integer.

To find the number of factors of 130 (factors 1 and 130 included)**

1) Do the prime factorization of 130: $$2^1*5^1*13^1$$

2) Take the exponents of each prime factor. Add 1 to each.

Exponents are 1, 1, 1
Add 1 to each, thus: 2, 2, 2

3) Multiply those numbers

2 * 2 * 2 = 8
There are 8 factors of 130*

*Factors of 130 are
1, 2, 5, 10, 13, 26, 65, 130
130 divided by any of them yields an integer

**The theory, from Bunuel , SEE THIS POST, scroll down to NUMBER OF FACTORS is, verbatim

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.
CEO
Joined: 11 Sep 2015
Posts: 3243
Re: For how many positive integer x is 130/x an integer?  [#permalink]

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13 Oct 2017, 06:03
Top Contributor
Bunuel wrote:
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

This question is asking us "How many positive divisors does 130 have?"

We can solve this using genxer123's technique or we can just list the divisors IN PAIRS and then count them. We get:
1 and 130
2 and 65
5 and 26
10 and 13

TOTAL = 8 divisors

Cheers,
Brent
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Re: For how many positive integer x is 130/x an integer?  [#permalink]

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16 Oct 2017, 16:08
Bunuel wrote:
For how many positive integer x is 130/x an integer?

(A) 8
(B) 7
(C) 6
(D) 5
(E) 3

If x is a positive integer and 130/x is an integer, then x must be a factor of 130. Thus, we need to determine the number of factors of 130. To determine the number of factors, we break down 130 into its prime factorization, add 1 to the exponent of each unique prime, and then multiply. Thus:

130 = 10 x 13 = 2^1 x 5^1 x 13^1

Thus, 130 has (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8 total factors.

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Re: For how many positive integer x is 130/x an integer? &nbs [#permalink] 16 Oct 2017, 16:08
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