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For how many positive integer x is 130/x an integer?
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22 May 2017, 06:26

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Bunuel wrote:

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

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

In order for 130/x to be an integer, x must be a divisor of 130

So, the question is really asking us to determine the number of positive divisors of 130. To do this, we can use the following rule: 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

Now onto the question....... 130 = (2)(5)(13) = (2^1)(5^1)(13^1) So, the number of positive divisors of 130 = (1+1)(1+1)(1+1) =(2)(2)(2) = 8

Re: For how many positive integer x is 130/x an integer?
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23 May 2017, 01:25

It is a direct question of finding no. of factors. Factorization of any integer = a^p * b^q * c^r (where a,b,c are prime factors and p,q,r are powers of a,b,c respectively). So on factorization of 130, we get 2^1 * 5^1 * 13^1 No. of factors = (p+1)*(q+1)*(r+1) = 2*2*2 = 8.

Re: For how many positive integer x is 130/x an integer?
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25 May 2017, 15:50

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 factor of 130, then 130/x is an integer; thus, we need to determine the number of factors that 130 has.

To do so, we can use the rule in which we prime factorize 130, then add 1 to each exponent of each unique prime factor and then multiply those values together.

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

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

Answer: A
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Re: For how many positive integer x is 130/x an integer?
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01 Oct 2018, 08:05

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