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GMAT Prep Question - Function

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ZT: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Tue Feb 21, 2012 4:40 pm

by ZT » Wed Apr 04, 2012 10:07 am

Please help solve the following. I'm having a hard time understanding what's actually being asked. Your help to explain will be much appreciated!

The function f is defined for all postive 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

Last edited by ZT on Thu Apr 05, 2012 6:38 am, edited 1 time in total.

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Source: Beat The GMAT — Problem Solving | Wed Apr 04, 2012 10:07 am

Pharo: Senior | Next Rank: 100 Posts; Posts: 61; Joined: Mon Jan 18, 2010 11:09 pm; Thanked: 8 times

by Pharo » Wed Apr 04, 2012 1:01 pm

Hey dude! I believe the question is missing a vital part :

The function f is defined for all postive 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) =

without "1"; the answer would be 0

Let me give you some output of the formula they have given us.

f(n) = (number of positive integers less than n) AND (has no positive factor in common with n other than 1)
f(3) = 2; what are the positive integers less than 3? They are 1 and 2. And Do they have a positive factor in common with n other than 1? Nope. So the output of the function is 2. (i.e the count of the numbers that satisfy both statements)
f(5) = 4; What are the positive integers less than 5? They are 1,2,3,4. Do they have a positive factor in common with n other than 1? Nope. So the output of the function is 4. (i.e. the count of the numbers that satisfy both statements)

And you realized it? I was giving you prime numbers as n

-- I gave you n=3; f(n) turned out to be 2. I said n = 5; f(n) turned out to be 4. (i.e f(n)= n-1 IF n is prime). Hence the answer should be A

The proof of the above would be this :

A prime number has no other divisor than 1 and itself. So any positive integer less than it will satisfy the "has no positive factor in common other than 1" clause. And how many positive integers are there less than it? Whatever the value is minus 1

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neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 amFollowed by: 52 members; Location: Utrecht, The Netherlands; Thanked: 318 times

by neelgandham » Wed Apr 04, 2012 1:02 pm

The function f is defined for all postive 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 nother than . 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

If p = 3, f(p) = 2 (Number : 2,1) Now let's check

A)p-1, p = 3, f(p) = 2 May be
B)p-2, p = 3, f(p) = 1, Oops, Incorrect
C)(p+1)/2, p = 3, f(p) = 2 May be
D)(p-1)/2, p = 3, f(p) = 1, Oops, Incorrect
E)2, May be

Now we have options B) and D) to choose
If p = 5, f(p) = 4 (Numbers: 1,2,3,4).

A)p-1, p = 5, f(p) = 4 May be
C)(p+1)/2, p = 5, f(p) = 3 Oops, Incorrect
E)2, Oops, Incorrect

Answer A

Edit: Corrected!

Anil Gandham
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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Wed Apr 04, 2012 6:44 pm

ZT wrote:Please help solve the following. I'm having a hard time understanding what's actually being asked. Your help to explain will be much appreciated!

The function f is defined for all postive 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 nother than . 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

Tricky solution:

Let us take p = 2 (smallest prime)
Now number of positive integers less than p and has no common factor with p other than 1 is 1. So f(2) = 1

Only option A satisfies this result.

Mathematical Approach:

Note that a prime number will have common factors other than 1 only with its multiples like pÂ², pÂ³ etc. As p is always greater than 1, all multiples of p are greater than p. Hence, none of the integers less than p will have any common factor with p.

Thus, f(p) = Number of positive integers less than p = (p - 1)

The correct answer is A.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
1-800-566-4043 (USA)

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ZT: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Tue Feb 21, 2012 4:40 pm

by ZT » Thu Apr 05, 2012 6:37 am

Whoa, thank you for the speedy and awesome responses, guys! It all makes sense now.

Pharo wrote:Hey dude! I believe the question is missing a vital part :

The function f is defined for all postive 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) =
without "1"; the answer would be 0

Great catch, Pharo. I've edited my original post to reflect the correct question. Hopefully it will help others who come across a similar problem.

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