Determine the common period

Question

Let $V = \mathrm{Vect}( e^{ 4 \pi it } , e^{ 5 \pi i t} , e^{ 6 \pi i t} ) $ and $f \in V$.

I want to determine the common period between the functions : $t \mapsto e^{ 4 \pi it } $, $t \mapsto e^{ 5 \pi i t }$ and $t \mapsto e^{ 6 \pi i t}$ ; i.e the period of $f \in V$.

We have

• $ t \mapsto e^{ 4 \pi i t} $ is periodic with period $\frac{1}{2}$
• $ t \mapsto e^{ 5 \pi i t} $ is periodic with period $\frac{2}{5}$
• $ t \mapsto e^{ 6 \pi i t} $ is periodic with period $\frac{1}{3}$

and I don't know how to pursue it. Is there a particular method to found a common period? I need a hint.

Thank you

Answer 1

Hint: You can use that

$$\operatorname{lcm}\biggl(\frac pq,\frac rs\biggr)=\frac{\operatorname{lcm}(p,r)}{\gcd(q,s)}.$$

Answer 2

The periods are $\frac{1}{2}, \frac{2}{5},\frac{1}{3}$.

Since $\operatorname{lcm}(2,5,3)=30$, we can write these periods as $\frac{15}{30}, \frac{12}{30},\frac{10}{30}$.

Since $\operatorname{lcm}(15,12,10)=60$, the lcm of the periods is $\frac{60}{30}=2$.