The function F is defined for all positive integers n by the following

If not the wording the question wouldn't be as tough as it is now. The GMAT often hides some simple concept in complicated way of delivering it.

This question for instance basically asks: how many positive integers are less than given prime number p which have no common factor with p except 1.

Well as p is a prime, all positive numbers less than p have no common factors with p (except common factor 1). So there would be p-1 such numbers (as we are looking number of integers less than p).

For example: if p=7 how many numbers are less than 7 having no common factors with 7: 1, 2, 3, 4, 5, 6 --> 7-1=6.

Answer: A.

Hope it's clear.

But it says no other factors in common with n other than 1, why do we have to include 1 then? I thought since 1 is a factor of 1 itself and p, we cannot include it.

Each positive integer should have no factor common with n except 1.
1 also has only a single factor i.e. 1 common with p. So we do include 1.

Okay let's see what happens here.

It took me 20 seconds to understand what the question was precisely asking for. But this is the most important step; do not attempt anything if you don't understand throughly the question.

What I figured is that the definition of "relatively prime" was pretty close to the description assigned to the question.

Two different numbers are said to be relatively prime whenever their GCF=1.

Let's pick a random example: how many numbers less than 105 are relatively prime to 105?

105=3(5)7 then the total number of relatively primes will be: 105(1-1/3)(1-1/5)(1-1/7)=48

Let's apply the same logic to our question and consider a random prime number: p(1-1/p)= p-1 which turns out to be the correct answer.

well damn...the wording is indeed confusing..as I was thinking that f(n) is the sum of the all numbers..

In such questions, it is advisable to take an example to figure out what the question is saying.

"The function f is defined for all the positive integers n by the following rule:"

We are looking at all positive integers so say n is 3.


"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 . "

Positive integers less than n -> 1, 2
Both do not have a factor in common with 3.
So f(n) = 2 (number of integers which have nothing in common with n except 1)

"if p is a prime number then f(p)?"

p must be a prime number. Our previous example was a prime number. Let's take another say 5.
Positive integers less than 5 -> 1, 2,3, 4
All 4 integers will have no factor in common with 5 because 5 is prime.
f(5) = 4

This will be the case with all prime numbers. Since a prime has no factor in common (except 1) with all positive integers less than it,
f(p) = p - 1

Answer (A)

Hi All,

This question is meant to ask: "If P is a prime number, then f(P)= ?" This question can be solved by TESTing VALUES.

Let's TEST N=7. The f(7) = all the positive integers less than 7 that have no factor in common with 7 except for 1.

THAT list is 1, 2, 3, 4, 5, 6 = 6 terms.

Thus, we're looking for an answer that equals 6 when we plug N=7 into it. There's only one answer that matches...

Final Answer:

GMAT assassins aren't born, they're made,
Rich

Since p can be ANY PRIME NUMBER, let p=2.
In this case:
f(p) = f(2) = the number of positive integers less than 2 that have no factor in common with 2.
Since only ONE positive integer is less than 2, f(2) = 1.
The correct answer must yield 1 when p=2.
Only A works:
p-1 = 2-1 = 1

We can see that, in other words, f(n) is the number of positive integers less than n that are relatively prime to n. If p is a prime, then any positive integer less than p will be relatively prime to p. For example, if p = 7, then f(7) = 6 since 1, 2, 3, 4, 5, and 6 are all relatively prime to 7. Therefore, f(p) = p - 1.

Answer: A

I read that the GMAT has been doing this lately.


They take actual theorems or formula from areas of math outside the scope of the test, define them, and use them as a “function” question.

The function described in the problem is called the “Euler Number”: how many Integers less than N are Co-Prime with N (I.e., do not share any factors).

For instance, to find the Euler number of 25, you would pull out all the integers less than 25 that share any prime factors with 25.

Euler number of 25 = 25 * (1 - (1/5) = 25 * 4/5 = 20

There are 20 numbers less than 25 that do not share any prime factors with 25—- only 1

Note: 1 is co-prime with every integer

For any Prime number that is only divisible by 1 and itself (P) ———> every number less than P will be Co-Prime with P

P - 1

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The function F is defined for all positive integers n by the following rule: f(n) is the number of positive integer 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

Find a pattern.

p is prime.

p = 2 --> f(2) = 1 (Why? 1 is the only factor that is POSITIVE and LESS THAN 2)
p = 3 --> f(3) = 2 (Factors: 1, 2)
p = 5 --> f(5) = 4 (1,2,3,4)
p = 7 --> f(7) = 6 (1,2,3,4,5,6)

Notice that the number of factors is always less than the prime number itself

p - 1

A.

The question currently says "If p is any number" instead of "If p is any prime number".

Fixed the typo. Thank you!