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let prime numbers be 3,4, then there are 4 possible divisors (1,3,4,12) but if the prime numbers are 1,2, then there are only 2 possible divisors (1,2)

let prime numbers be 3,4, then there are 4 possible divisors (1,3,4,12) but if the prime numbers are 1,2, then there are only 2 possible divisors (1,2)

Re: If n is an integer, n is divisible by how many positive [#permalink]
07 Feb 2014, 17:37

If n is an integer, n is divisible by how many positive integers? -> # factors of n????

i) n is the product of two prime numbers -> n = p^1 * q^1, where p and q are prime; number of factors of n = 4. ii) n and 2^3 are each divisible by the same number of positive integers -> 2^3 has 4 factors => n has 4 factors.

Re: If n is an integer, then n is divisible by how many positive [#permalink]
08 Feb 2014, 01:05

Expert's post

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. For more on number properties check: math-number-theory-88376.html

BACK TO THE ORIGINAL QUESTION:

If n is an integer, then n is divisible by how many positive integers?

(1) n is the product of two different prime numbers --> n=ab, where a and b are primes, so # of factors is (1+1)(1+1)=4. Sufficient.

(2) n and 2^3 are each divisible by the same number of positive integers --> 2^3 has 4 different positive factors (1, 2, 4, and 8) so n has also 4. Sufficient.