Arithmetic operations on periods of functions

Question

If $ f(x) $ has a period $ 7 $ and $ g(x) $ has a period $ 11 $, then the period of $ F(x) = f(x)g(\frac{x}{5}) - g(x)f(\frac{x}{3}) $.

Answer 1

$F(x)=f(x)g(x/5)-g(x)f(x/3)$ has period $11\times 15\times 7 = 1155$

Note that $$ F(x+1155)=$$

$$ f(x+11\times 15\times 7 )g(x/5+11\times 3\times 7)-g(x+11\times 15\times 7)f(x/3+11\times 5\times 7) =$$

$$f(x)g(x/5)-g(x)f(x/3)=F(x)$$