Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1992 AIME Problems/Problem 15"

 
 
(13 intermediate revisions by 10 users not shown)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
 +
Define a positive integer <math>n^{}_{}</math> to be a factorial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal representation of <math>m!</math> ends with exactly <math>n</math> zeroes. How many positive integers less than <math>1992</math> are not factorial tails?
  
 
== Solution ==
 
== Solution ==
 +
Let the number of zeros at the end of <math>m!</math> be <math>f(m)</math>. We have <math>f(m) = \left\lfloor \frac{m}{5} \right\rfloor + \left\lfloor \frac{m}{25} \right\rfloor + \left\lfloor \frac{m}{125} \right\rfloor + \left\lfloor \frac{m}{625} \right\rfloor + \left\lfloor \frac{m}{3125} \right\rfloor + \cdots</math>.
 +
 +
Note that if <math>m</math> is a multiple of <math>5</math>, <math>f(m) = f(m+1) = f(m+2) = f(m+3) = f(m+4)</math>.
 +
 +
Since <math>f(m) \le \frac{m}{5} + \frac{m}{25} + \frac{m}{125} + \cdots  = \frac{m}{4}</math>, a value of <math>m</math> such that <math>f(m) = 1991</math> is greater than <math>7964</math>. Testing values greater than this yields <math>f(7975)=1991</math>.
 +
 +
There are <math>\frac{7975}{5} = 1595</math> distinct positive integers, <math>f(m)</math>, less than <math>1992</math>. Thus, there are <math>1991-1595 = \boxed{396}</math> positive integers less than <math>1992</math> that are not factorial tails.
 +
 +
== Solution 2 ==
 +
 +
After testing various values of <math>m</math> in <math>f(m)</math> of solution 1 to determine <math>m</math> for which <math>f(m) = 1992</math>, we find that <math>m \in \{7980, 7981, 7982, 7983, 7984\}</math>. WLOG, we select <math>7980</math>. Furthermore, note that every time <math>k</math> reaches a multiple of <math>25</math>, <math>k!</math> will gain two or more additional factors of <math>5</math> and will thus skip one or more numbers.
 +
 +
With this logic, we realize that the desired quantity is simply <math>\left \lfloor \frac{7980}{25} \right \rfloor + \left \lfloor \frac{7980}{125} \right \rfloor \cdots</math>, where the first term accounts for every time <math>1</math> number is skipped, the second term accounts for each time <math>2</math> numbers are skipped, and so on. Evaluating this gives us <math>319 + 63 + 12 + 2 = \boxed{396}</math>. - Spacesam(edited by srisainandan6)
  
 
== See also ==
 
== See also ==
−
* [[1992 AIME Problems]]
+
{{AIME box|year=1992|num-b=14|after=Last Question}}
 +
 
 +
[[Category:Intermediate Number Theory Problems]]
 +
{{MAA Notice}}

Latest revision as of 00:54, 2 October 2020

Problem

Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?

Solution

Let the number of zeros at the end of $m!$ be $f(m)$. We have $f(m) = \left\lfloor \frac{m}{5} \right\rfloor + \left\lfloor \frac{m}{25} \right\rfloor + \left\lfloor \frac{m}{125} \right\rfloor + \left\lfloor \frac{m}{625} \right\rfloor + \left\lfloor \frac{m}{3125} \right\rfloor + \cdots$.

Note that if $m$ is a multiple of $5$, $f(m) = f(m+1) = f(m+2) = f(m+3) = f(m+4)$.

Since $f(m) \le \frac{m}{5} + \frac{m}{25} + \frac{m}{125} + \cdots = \frac{m}{4}$, a value of $m$ such that $f(m) = 1991$ is greater than $7964$. Testing values greater than this yields $f(7975)=1991$.

There are $\frac{7975}{5} = 1595$ distinct positive integers, $f(m)$, less than $1992$. Thus, there are $1991-1595 = \boxed{396}$ positive integers less than $1992$ that are not factorial tails.

Solution 2

After testing various values of $m$ in $f(m)$ of solution 1 to determine $m$ for which $f(m) = 1992$, we find that $m \in \{7980, 7981, 7982, 7983, 7984\}$. WLOG, we select $7980$. Furthermore, note that every time $k$ reaches a multiple of $25$, $k!$ will gain two or more additional factors of $5$ and will thus skip one or more numbers.

With this logic, we realize that the desired quantity is simply $\left \lfloor \frac{7980}{25} \right \rfloor + \left \lfloor \frac{7980}{125} \right \rfloor \cdots$, where the first term accounts for every time $1$ number is skipped, the second term accounts for each time $2$ numbers are skipped, and so on. Evaluating this gives us $319 + 63 + 12 + 2 = \boxed{396}$. - Spacesam(edited by srisainandan6)

See also

1992 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1992_AIME_Problems/Problem_15&oldid=134470"