Value of N!

[Ahmed MS]

Q: If n is a positive integer less than 200 and 14n�60 is also an integer, then n has how many different positive prime factors?

a. 2
b. 3
c. 5
d. 6
e. 8

The answer is B, but I failed to understand. Apology is there any error in the question as I have found in the web.

Cheers!

[Frankenstein]

Hi,
n < 200
14n/60 is integer i.e. 7n/30 is integer. So, n must be a multiple of 30.
Lets say n =30*k. As n<200, k <= 6
30 = 2*3*5 (prime factors are 2,3,5)
n= 30*k, where k<=6 will have only these 3 prime factors as k<7(next distinct prime factor).

Hence, B

[winniethepooh]

The answer B is correct and the value of N=30.
Consider n<200
Also, 14 * N/ 60 = an integer

Now, 14 * multiples of 5s or multiples of 10s alone give you round figures,example 14*5 = 70 or 14*10 =140.....
You have to find the value of N such that if multiplied by 14 is divisible by 60..
You can go on to find 14 * 5 or 10 or 15 or 20 or 25 or 30 - Stop.
!4 * 30 is divisible by 60..
30 is divisible by 2,3,5(three prime factors)
Hensce, answer =3

[sunilramu]

N < 200
14*N / 60 is an integer.

=> 2.7.N/2.2.3.5 is an integer
=> N should at least have 2.3.5 in it.
=> Now if you add next prime factor(7) to the list. It would make n > 200 (2.3.5.7 = 210)

So N has 3 distinct positive prime factors.