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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]

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07 Feb 2014, 18:37

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.

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.