What is the minimum period of the discrete sequence \(e^j\dfrac{2\pi n}{4} + e^j\dfrac{2\pi n}{6}\)

Fundamental period: 

The smallest possible value of a period is called the fundamental period of a given sequence.

\(let \ \rm x[n]=e^{j\Omega n}\) than fundamental period (N) of x[n] is given as,

\(N={\frac {2\pi}{\Omega}}\times k\)

where k is an integer.

Given sequence is:

 \(x[n]=e^j\dfrac{2\pi n}{4} + e^j\dfrac{2\pi n}{6}\)

then period for \(e^j\dfrac{2\pi n}{4} \) is 

\(N _1=(\frac{2 \pi}{\frac{2\pi}{4}})\times k\implies4\)

then the period for \(e^j\dfrac{2\pi n}{6} \) is 

\(N _1=(\frac{2 \pi}{\frac{2\pi}{6}})\times k\implies6\)

overall period for x[n] is given as 

N = LCM (N₁,N₂)

N = LCM (4,6)

N = 12