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*
Answer A
*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 IntegerFirst 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.
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