Prime factors

[tanyajoseph]

If n is a postive integer less than 200 and 14n/60 is an integer, then n has how many different positive prime factors.
2
3
5
6
8

Ans is 3.
I assumed it would be 3 as, 60 is a factor of 2,2 and 5.
14 is a factor of 2 and 7. So cancelling common factor 2 and 5 would remain...so should it not be 2 prime factors (2,5) HOw does the count go to 3???

[ri2007]

You know 14n/60 is an integer

so 7n/30 remainder should be 0

So N should have all the prime numbers in 30 = 2*3*5

Now can n have any other prime number - next smallest prime no is 7, but 7 * 30 = 210

since n is less than 200 n can only have 3 prime nos i.e 2, 3 & 5

[samirpandeyit62]

I agree wih RI

14n/60 is 7n/30

i.e 7n/ (3*2*5)

now here 7 is not divisible by either of these 3 prime factors (3,2,5)

so they must divide n evenly to get and integer as result of this fraction

hence 3,2,5 are prime factors of n

now we can say that prime factors like 7, 11 etc can also be prime factors of n

but 3*2*5*7 >210 so 7,11 etc cannot be prime factors of n

hence ans 3