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 "2015 AMC 10B Problems/Problem 23"

(→‎Solution 4)
(→‎Solution 4)
Line 44: Line 44:
 
<cmath>3(\lfloor\frac{5m+k}{5}\rfloor+\lfloor\frac{5m+k}{25}\rfloor)=\lfloor\frac{10m+2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor</cmath>
 
<cmath>3(\lfloor\frac{5m+k}{5}\rfloor+\lfloor\frac{5m+k}{25}\rfloor)=\lfloor\frac{10m+2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor</cmath>
 
<cmath>3m+3\lfloor\frac{5m+k}{25}\rfloor=2m+\lfloor\frac{2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor</cmath>
 
<cmath>3m+3\lfloor\frac{5m+k}{25}\rfloor=2m+\lfloor\frac{2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor</cmath>
 +
<cmath>m+3\lfloor\frac{5m+k}{25}\rfloor=\lfloor\frac{2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor</cmath>
 +
To minimize <math>n</math>, we let <math>\lfloor\frac{5m+k}{25}\rfloor=\lfloor\frac{10m+2k}{25}\rfloor=0</math>
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2015|ab=B|num-b=22|num-a=24}}
 
{{AMC10 box|year=2015|ab=B|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 13:58, 9 August 2019

Problem

Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$?

$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

Solution

Solution 1

A trailing zero requires a factor of two and a factor of five. Since factors of two occur more often than factors of five, we can focus on the factors of five. We make a chart of how many trailing zeros factorials have:

\[\begin{array}{c|c|c|c|c|c|c} \mathrm{Factorial}&0!-4!&5!-9!&10!-14!&15!-19!&20!-24!&25!-29!\\\hline \mathrm{Zeros}&0&1&2&3&4&6 \end{array}\]

We first look at the case when $n!$ has $1$ zero and $(2n)!$ has $3$ zeros. If $n=5,6,7$, $(2n)!$ has only $2$ zeros. But for $n=8,9$, $(2n)!$ has $3$ zeros. Thus, $n=8$ and $n=9$ work.

Secondly, we look at the case when $n!$ has $2$ zeros and $(2n)!$ has $6$ zeros. If $n=10,11,12$, $(2n)!$ has only $4$ zeros. But for $n=13,14$, $(2n)!$ has $6$ zeros. Thus, the smallest four values of $n$ that work are $n=8,9,13,14$, which sum to $44$. The sum of the digits of $44$ is $\boxed{\mathbf{(B)\ }8}$

Solution 2

By Legendre's Formula and the information given, we have that $3\left(\left\lfloor{\frac{n}{5}}\right\rfloor+\left\lfloor{\frac{n}{25}}\right\rfloor\right)=\left\lfloor{\frac{2n}{5}}\right\rfloor+\left\lfloor{\frac{2n}{25}}\right\rfloor$.

Trivially, it is obvious that $n<100$ as there is no way that if $n>100$, $n!$ would have $3$ times as many zeroes as $(2n)!$.

First, let's plug in the number $5$ We get that $3(1)=1$, which is obviously not true. Hence, $n>5$

After several attempts, we realize that the RHS needs $1$ to $2$ more "extra" zeroes than the LHS. Hence, $n$ is greater than a multiple of $5$.

Very quickly, we find that the least $4 n's$ are $8,9,13,14$.

$8+9+13+14=17+27=44\implies 4+4=8\implies\boxed{B}$.

Solution 3 (Bashing)

We notice that for a $0$ to be at the end of a factorial, one multiple of five must be there. Therefore, it is intuitive to start at $5$ and work up. If you bash enough you get $8$, $9$, $13$, and $14$. Going any higher will give too many zeros, and then we can stop going higher. $8+9+13+14=17+27=44\implies 4+4=8\implies\boxed{\textbf{(B) }8}$.


Solution 4

Let $n=5m+k$ for some natural numbers $m$, $k$ such that $k\in\{0,1,2,3,4\}$. Notice that $n<5^3=125$. Thus \[3(\lfloor\frac{n}{5}\rfloor+\lfloor\frac{n}{25}\rfloor)=\lfloor\frac{2n}{5}\rfloor+\lfloor\frac{2n}{25}\rfloor+\lfloor\frac{2n}{125}\rfloor\] For smaller $n$, we temporarily let $\lfloor\frac{2n}{125}\rfloor=0$ \[3(\lfloor\frac{n}{5}\rfloor+\lfloor\frac{n}{25}\rfloor)=\lfloor\frac{2n}{5}\rfloor+\lfloor\frac{2n}{25}\rfloor\] \[3(\lfloor\frac{5m+k}{5}\rfloor+\lfloor\frac{5m+k}{25}\rfloor)=\lfloor\frac{2(5m+k)}{5}\rfloor+\lfloor\frac{2(5m+k)}{25}\rfloor\] \[3(\lfloor\frac{5m+k}{5}\rfloor+\lfloor\frac{5m+k}{25}\rfloor)=\lfloor\frac{10m+2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor\] \[3m+3\lfloor\frac{5m+k}{25}\rfloor=2m+\lfloor\frac{2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor\] \[m+3\lfloor\frac{5m+k}{25}\rfloor=\lfloor\frac{2k}{5}\rfloor+\lfloor\frac{10m+2k}{25}\rfloor\] To minimize $n$, we let $\lfloor\frac{5m+k}{25}\rfloor=\lfloor\frac{10m+2k}{25}\rfloor=0$

See Also

2015 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 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=2015_AMC_10B_Problems/Problem_23&oldid=108629"