Thonk02 wrote:A function is defined by the fule that F(N) equals the nmber of positive integeres less than N and has no factor common with N other than 1. If P is Prime then, F(P) =?
A. P-1
B. P-2
C. P+1/2
D. P-1/2
E. 2
Answer is A. any help on how to solve this would be appreciative.
Let's start by understanding the question (always a good first step!).
F(N) is the number of positive intgers less than N with no factors in common with N other than 1.
For example, if N=4, then F(N)=2, since of the positive integers less than 4 (1, 2 and 3), 2 has a factor in common with 4 ("2") and 1 and 3 have no non-1 factors in common with 4.
In the actual question, we're told that P is a prime number and we want the value of F(P).
Like many GMAT problems, there are two effective ways to solve: using our knowledge of mathematical principles or getting our hands dirty and picking numbers.
Let's start by picking numbers.
If P=3, then F(P) = 2, since neither 1 nor 2 have any non-1 factors in common with 3.
Plugging 3 into the answer choices, both A and E give us the correct answer ("2").
If P=5, then F(P) = 4, since none of 1, 2, 3 and 4 have any non-1 factors in common with 5.
Plugging 5 into A and E, only A is still correct: choose A.
* * *
We can also use our knowledge of primes to solve.
Prime numbers, by definition, have only 2 factors: 1 and themselves. Therefore, all numbers below a given prime will have no non-1 factors in common with that prime.
Since there are (P-1) positive integers less than P, and since none of them have non-1 factors in common with P, (P-1) is the correct answer to the question: choose A.
