Solution

Find the value of $\varphi(75600)$.


Note that the prime factorization of $75600$ is given by $$75600 = 2^4 \cdot 3^3 \cdot 5^2 \cdot 7$$ Thus, $$\begin{array}{rcl} \varphi(75600) &=& \varphi(2^4) \varphi(3^3) \varphi(5^2) \varphi(7)\\ &=& (2^4-2^3)(3^3-3^2)(5^2-5)(7-1)\\ &=& 8 \cdot 18 \cdot 20 \cdot 6\\ &=& 17280 \end{array}$$