“I can’t seem to figure out how the GMAT arrived at the answer to this question on a GMATPrep test . Can you assist?”
Q: The function f is defined for all positive integers n by the following rule: f(n) is the number of positive integers each of which is less than n and has no positive factor in common with n other than 1. If p is any prime number, then f(p) =
A) p – 1
B) p – 2
C) (p+1)/2
D) (p-1)/2
E) 2
The answer is p-1. Why is the answer not 2?
Thanks,
GMAT Pill Response
Just pick any prime number like 7 or 11.
Say 7. f(7) = # of positive integers less than 7. Well, there are 6 positive numbers less than 7: 1, 2, 3, 4, 5, and 6.
But it also has to have no positive factor in common with 7 other than 1.
So just go through the list: 1, 2, 3, 4, 5, and 6—–compare them each with 7 and 1 like so:
7 and 6: Do they share no positive factor other than 1? Yes.
7 and 5: Do they share no positive factor other than 1? Yes.
7 and 4: Do they share no positive factor other than 1? Yes.
7 and 3: Do they share no positive factor other than 1? Yes.
7 and 2: Do they share no positive factor other than 1? Yes.
7 and 1: Do they share no positive factor other than 1? Yes. The only factor that 7 and 1 have in common is 1. So yes, this counts as one also.
And the above up and basically all the numbers work between 1-6 so since 6 is one less than 7, then the answer is 1 less than n.
So answer is p-1. Hope that helps!
Zeke
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