Question

What is the greatest power of 5 which can divide $80!$ exactly?

• A. 16
• B. 20
• C. 19
• D. None of these

Hint:

Use Legendre’s formula to find highest power of a prime dividing factorial: successively divide by $p$, $p^2$, $p^3$...

Answer 1

The Correct Option is C

To find highest power of a prime $p$ in $n!$, use: \[ \left\lfloor \frac{n}{p} \right\rfloor + \left\lfloor \frac{n}{p^2} \right\rfloor + \left\lfloor \frac{n}{p^3} \right\rfloor + \ldots \] Apply for $p = 5$, $n = 80$ \[ \left\lfloor \frac{80}{5} \right\rfloor = 16,\quad \left\lfloor \frac{80}{25} \right\rfloor = 3,\quad \left\lfloor \frac{80}{125} \right\rfloor = 0 \] Total power of 5 = $16 + 3 = \boxed{19}$