Euler is not Haunting!

Solution:

$\phi(n)=n-1$ is true for all prime numbers.

$\phi(n)=n\left(1-\frac{1}{p_1}\right)(\cdots)\left(1-\frac{1}{p_k}\right)$

where $p_1,p_2,p_k$ are distinct primes dividing $n$ . Since a prime number is divisible by itself and 1 only so $p_1=n$

$\begin{aligned}\phi(n)&=n\left(1-\frac{1}{n}\right) \\ &=n\left(\frac{n-1}{n}\right)\\ \phi(n)&=n-1\\ \\ n<20 \end{aligned}$

So,prime numbers less than 20 are $2,3,5,7,11,13,17,19$

$\therefore$ Number of integers less than 20 which satisfy the above condition is $\boxed{8}$