Xbond wrote:Hi there,
Could you explain the concept of this DS in simplest way
If n is a positive integer less than 200 and 14n/60 is an integer, then n has how many different
positive prime factors ?
a)Two
b)Three
c)Five
d)Six
e)Eight
We know that 14n/60 is an integer; therefore, 14n is a multiple of 60.
For 14n to be a multiple of 60, 14n must contain (at a minimum) the same primes as does 60. Let's start by breaking down 60:
60 = 2*30 = 2*2*15 = 2*2*3*5
The "14" part of 14n contains a "2". Therefore, n must be responsible for the remaining primes of 60: 2, 3 and 5.
Therefore, the minimum possible value for n is 2*3*5 = 30. At this point there are two ways we can finish the question.
First, we can use some logic. The answer choices are numbers and only one of them can be correct. Since we already know that n
could have 3 different primes, 3
must be the correct answer to the question.
Alternatively, we can use the info in the question stem. We know that 0<n<200. The next smallest prime is 7; since 30*7 = 210, we cannot add 7 to the prime factors of n without violating the rule. Therefore, n must have only 3 different prime factors.
Note that "different" is a key word in the question; n could also be 60, 90, 120, 150 or 180 (i.e. any multiple of 30 that's less than 200); however, all of these numbers have the same 3 distinct primes, so all of them generate the same answer to the question.
