The largest n N such that 3^n divides 50 !

Question

The largest $n\in N$ such that ${3}^{n}$ divides 50 ! is :

Accepted Answer

Using Legendre's formula, the exponent of 3 in $50!$ is

$$\left\lfloor\frac{50}{3}\right\rfloor+\left\lfloor\frac{50}{9}\right\rfloor+\left\lfloor\frac{50}{27}\right\rfloor+\left\lfloor\frac{50}{81}\right\rfloor+\cdots = 16+5+1+0=22$$

Solutions · 1