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 $f(p)=$

Correct answer marked below

p - 1

Correct

p - 2

(p + 1)/2

(p - 1)/2

2

You can see the correct answer above. Sign in for free to unlock the complete worked solution.

Get detailed analytics, unlock full explanations, and move up difficulty tiers as you practice.

Explanation

Create a free account to reveal the correct answer, see the step-by-step explanation, and start tracking your GMAT progress.

Comments

Prad18h ago

This is just Euler's totient function φ(n), right? For a prime p, all numbers 1 through p-1 are coprime to p, so f(p) = p - 1.

Sebastianjust now

Right, the key is that a prime has no factors other than 1 and itself, so nothing below it shares a factor.

AddyontherocksJun 5

Took me a second to parse "no positive factor in common other than 1." Once I realized it's just counting coprime integers, it clicked.

JasonJun 15

Same, the wording is a bit clunky but the concept is straightforward once you translate it.

Paul5d ago

Why is (p-1)/2 wrong? I keep second-guessing myself.

Johnjust now

Because it's not just the odds — every integer from 1 to p-1 is coprime to p, not half of them. Try p=7 and count.

FrancescoMay 29

Solid 655-level question. Concept is easy if you know totient, but the definition is written to make you slow down.

What this question tests

Your ability to evaluate and compose functions, including unusual defined operations.

Applying function operations in the wrong order or ignoring domain restrictions.

The function F is defined for all positive integers n by the following rule: f(n)f(n)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)=f(p)=f(p)=