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

Question

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

A. p - 1
B. p - 2
C. (p + 1)/2
D. (p - 1)/2
E. 2

Bunuel · Accepted Answer

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 confusing moment in this question is its wording. Basically question is: how many positive integers are less than given prime number p which has 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).

If we consider 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.

enigma123 · Answer

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

Guys - does this questions makes sense to anyone? I am struggling. Does it mean that:

F(n) is a list of positive integers. AM I right?

for e.g f(5) = 3,4.

I am stuck after this. Can someone please help?

Bunuel · Answer

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.

BN1989 · Answer

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.

KarishmaB · Answer

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.

gmat6nplus1 · Answer

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.

mvictor · Answer

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

KarishmaB · Answer

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)

EMPOWERgmatRichC · Answer

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:  A

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

GMATGuruNY · Answer

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

A

ScottTargetTestPrep · Answer

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

Fdambro294 · Answer

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

Posted from my mobile device

CEdward · Answer

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

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.

ManifestDreamMBA · Answer

p is any prime number then f(p), then number of positive integers each of which is less than p with no positive factor in common with p other than 1, will be all numbers but p
f(p) = p-1

For example is p = 5, then the numbers satisfying the condition are 1,2,3,4. f(5) = 4, i.e. 5-1 = 4

Answer A

Rishm · Answer

In such questions, better to assume P as 3 and 5 as options, for 3, you will have 2 numbers hence P-1, for 5 you will have 4 numbers hence p-1 & so on. Hence option A is the best option.

Usernamevisible · Answer

Bunuel  - why should 1 be counted in this solution for taking f(7) ? 
I am confused and marked p-2

Bunuel · Answer

“Other than 1” does not mean that 1 should be excluded from the count. It means that the number being counted must not have any common factor with p except 1.

So 1 qualifies: it is less than p, and its only common factor with p is 1.

egmat · Answer

Hi Usernamevisible,

Your count is almost right - the only thing tripping you up is what the phrase "other than 1" is actually doing.

Read the condition carefully

The rule isn't "exclude the number 1." It is: the integer you are counting must share no factor with p  except  1. In other words, the only factor it is allowed to have in common with p is 1 itself.

So when you test the number  1  against  7 :
- The factors of  1  are just {1}.
- The only factor 1 shares with 7 is  1 .
- That is exactly what the condition allows, so  1  qualifies and gets counted.

The phrase "other than 1" describes the  permitted  common factor, not a number to throw out. Every integer from  1  to  6  shares only the factor 1 with 7 (since 7 is prime), so all six count:

1, 2, 3, 4, 5, 6 gives f(7) =  6  = 7 - 1.

That is why the answer is p - 1, not p - 2. You would only get p - 2 if 1 really had to be dropped, but it does not.

Quick check to lock it in

Try f(5). The numbers less than 5 are 1, 2, 3, 4. Does 1 share any factor with 5 besides 1? No. So all four count, giving f(5) =  4  = 5 - 1. Same pattern for every prime.

Answer: A

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards