Let $\phi (n)$ be the cardinality of the set:

​$$\{a\mid1\leq a \leq n, gcd(a,n)=1\}$$

Where $\gcd (a,n)$ denotes the greatest common divisor of $a$ and $n$.

Which of the following is NOT true?​​

{a∣1≤a≤n,gcd⁡(a,n)=1}\{ a \mid 1 \leq a \leq n, \, \gcd(a, n) = 1 \}

Correct option is D

The Euler's totient function $\phi (n)$ counts the number of integers less than or equal to $n$ that are coprime with $n$. Let's analyze the options one by one.

Option (a): There exist infinitely many $n$ such that $\phi (n)>\phi (n+1)$.

• This is true. For certain values of $n$, the value of $\phi (n)$ decreases when moving to $n+1$, particularly for powers of primes or numbers with multiple prime factors.

Option (b): There exist infinitely many $n$ such that $\phi (n)<\phi (n+1)$.

• This is true. For some values of $n$, $\phi (n)$ increases when moving to $n+1$. This happens in cases when $n+1$ has fewer prime factors compared to $n$.

Option (c): There exists $N\in \mathbb{N}$ such that $N>2$ and for all $n>N$, $\phi (N)<\phi (n)$.

• This is true. There is a natural number $N$ such that for all $n>N$, $\phi (n)$ will eventually be greater than $\phi (N)$.

Option (d): The set $\{\phi (n)\mid n\in \mathbb{N}\}$ has finitely many limit points.

• This is not true. The Euler’s totient function $\phi (n)$ has infinitely many distinct values so $$\frac{\varphi(n)}{n}$$​ does not have finitely many limit points. This makes option (d) the false statement.

Conclusion:

The correct answer is (d) The set $\{\phi (n)\mid n\in \mathbb{N}\}$ has finitely many limit points.