If n is an integer, then n divisible by how many positive integers?Finding the Number of Factors of an Integer
(1) n is the product of two different prime numbers.
(2) n and 2^3 are each divisible by the same number of positive integers.
First make prime factorization of an integer n=a^p*b^q*c^r
, where a
, and c
are prime factors of n
, and r
are their powers.
The number of factors of n
will be expressed by the formula (p+1)(q+1)(r+1)
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
For more on number properties check: math-number-theory-88376.html
BACK TO THE ORIGINAL QUESTION:If n is an integer, then n 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.
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