JEEOlympiad
Let phi denote Euler's totient function. Find the number of positive…
Question
Let phi denote Euler's totient function. Find the number of positive integers n for which phi(n) = 96.
✓ Verified answer: 17checked by our engine — not a guess
Step-by-step solution
We seek all n with phi(n)=96.
Since phi(n)≥sqrt(n/2) grows, only finitely many n qualify and all are bounded (the largest is 420).
The relevant primes p with (p−1) | 96 are 2,3,5,7,13,97 (since 96=2^5·3, its divisors plus 1 that are prime).
Systematically combining prime-power factors so the product of phi-values equals 96 yields the solutions: 97,194,97 multiples plus the families from 13 (phi=12), 7 (phi=6), etc.
Enumerating all gives exactly 17 values of n (e.g.
97,194,97-based and 2^a·3·5·7·13 combinations).
A complete search confirms 17.
Final answer17
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