2006 AIME A Problems/Problem 3

Revision as of 14:35, 24 July 2006 by JBL (talk | contribs) (Solution: minor error + added internal links)

Problem

Let $\displaystyle P$ be the product of the first $\displaystyle 100$ positive odd integers. Find the largest integer $\displaystyle k$ such that $\displaystyle P$ is divisible by $\displaystyle 3^k .$

Solution

Note that the product of the first $\displaystyle 100$ positive odd integers can be written as $\displaystyle 1\cdot 3\cdot 5\cdot 7\cdots 195\cdot 197\cdot 199=\frac{1\cdot 2\cdots200}{2\cdot4\cdots200} = \frac{200!}{2^{100}\cdot 100!}$

Hence, we seek the number of threes in $\displaystyle 200!$ decreased by the number of threes in $\displaystyle 100!.$

There are

$\displaystyle \lfloor \frac{200}{3}\rfloor+\lfloor\frac{200}{9}\rfloor+\lfloor \frac{200}{27}\rfloor+\lfloor\frac{200}{81}\rfloor =66+22+7+2=97$

threes in $\displaystyle 200!$ and

$\displaystyle \lfloor \frac{100}{3}\rfloor+\lfloor\frac{100}{9}\rfloor+\lfloor \frac{100}{27}\rfloor+\lfloor\frac{100}{81}\rfloor=33+11+3+1=48$

threes in $\displaystyle 100!$

Therefore, we have a total of $\displaystyle 97-48=049$ threes.

For more information, see also prime factorizations of a factorial.

See also