0

Why is $$\text{lcm}\left(\frac{\pi}{5},\frac{\pi}{2}\right)=\pi$$ where the $10$ here represents the period of $2\cos(10t+1)-\sin(4t-1)$

where $\displaystyle\frac{\pi}{5}$ is the period of $2\cos(10t+1)$ and $\displaystyle\frac{\pi}{2}$ is the period of $-\sin(4t-1)$.

The question is not why the lcm but it's why does the lcm of these two give $\pi$.

XiChan
  • 161
  • Is your question why the LCM gives the period of the sum or why the LCM of $\pi/5,\pi/2$ is $\pi$? – Shubham Johri Mar 07 '21 at 18:22
  • I want to mark your answer as correct but MSE is telling me to wait 8 more minutes @ShubhamJohri Thank you for your answer I understood it :) – XiChan Mar 07 '21 at 18:23

2 Answers2

1

Suppose $n\pi/5=m\pi/2$ where $n,m\in\Bbb Z_{>0}$. For least multiple we need to minimize $m$. We get $n/m=5/2$ and the minimum $m$ that gives $n$ integral is $m=2$.

Shubham Johri
  • 17,659
1

$$LCM(\frac{a}{b},\frac{c}{d},\frac{e}{f})=\frac{LCM(a,c,e)}{HCF{(b,d,f)}}.$$

Z Ahmed
  • 43,235