# Which of the following is (are) true if f(n)=the number of distinct primes that divide n for n>1 (!)f(10)=f(12) (!!)If f(n)>1,then f(f(n))<f(n) (!!!)If f(n) = 3, then n<1000 A)All...

Which of the following is (are) true if f(n)=the number of distinct primes that divide n for n>1

(!)f(10)=f(12)

(!!)If f(n)>1,then f(f(n))<f(n)

(!!!)If f(n) = 3, then n<1000

A)All

B)(!)and (!!)only

C)(!) only

D)(!) and (!!!) only

E)None

### 1 Answer | Add Yours

`f(10)=f(2xx5)=2 . f(12)=f((2^2)xx3)=2` .So `f(10)=f(12)` .Clearly `f(n)<n` .Then `f(f(n))<f(n)` .So (!!) is true .But (!!!) is not true as we could let n be the product of three distinct primes greater than `1 000` .Then ` n>1 000` but` f(n)=3`