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 "2022 AMC 10A Problems/Problem 19"

m
m
Line 41: Line 41:
 
~sirswagger21
 
~sirswagger21
  
 +
== Solution 3 ==
 
Using Wolstenholmes' Theorem, we can rewrite <math>1 + \frac{1}{2} \dots + \frac{1}{16}</math> as <math>\frac{17^2 n}{(17 - 1)!} = \frac{17^2 n}{16!}</math> (for some <math>n \in \mathbb{Z}</math>). Adding the <math>\frac{1}{17}</math> to <math>\frac{17^2 n}{16!}</math>, we get <math>\frac{17^3 n + 16!}{17!}</math>.  
 
Using Wolstenholmes' Theorem, we can rewrite <math>1 + \frac{1}{2} \dots + \frac{1}{16}</math> as <math>\frac{17^2 n}{(17 - 1)!} = \frac{17^2 n}{16!}</math> (for some <math>n \in \mathbb{Z}</math>). Adding the <math>\frac{1}{17}</math> to <math>\frac{17^2 n}{16!}</math>, we get <math>\frac{17^3 n + 16!}{17!}</math>.  
  
 
Now we have <math>\frac{17^3 n + 16!}{17!} = \frac{h}{L_{17}}</math> and we want <math>h \pmod{17}</math>. We find that <math>\frac{L_{17}(17^3 n + 16!)}{17!} = \frac{L_{16}(17^3 n + 16!)}{16!} = h</math>. Taking <math>\pmod{17}</math> and multiplying, we get <math>L_{16}(17^3 n + 16!) \equiv 16! \cdot h \pmod{17}</math>.
 
Now we have <math>\frac{17^3 n + 16!}{17!} = \frac{h}{L_{17}}</math> and we want <math>h \pmod{17}</math>. We find that <math>\frac{L_{17}(17^3 n + 16!)}{17!} = \frac{L_{16}(17^3 n + 16!)}{16!} = h</math>. Taking <math>\pmod{17}</math> and multiplying, we get <math>L_{16}(17^3 n + 16!) \equiv 16! \cdot h \pmod{17}</math>.
  
−
 
−
==Solution 3==
 
 
Applying Wilson's Theorem on <math>16!</math> and reducing, we simplify the congruence to <math>L_{16}(0 - 1) \equiv -L_{16} \equiv -h \pmod{17}</math>. Now we proceed with Solution 1 and find that <math>L_{16} \equiv 5 \pmod{17}</math>, so our answer is <math>\boxed{\textbf{(C) }5}</math>.   
 
Applying Wilson's Theorem on <math>16!</math> and reducing, we simplify the congruence to <math>L_{16}(0 - 1) \equiv -L_{16} \equiv -h \pmod{17}</math>. Now we proceed with Solution 1 and find that <math>L_{16} \equiv 5 \pmod{17}</math>, so our answer is <math>\boxed{\textbf{(C) }5}</math>.   
  

Revision as of 18:46, 10 December 2022

Problem

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that \[\frac{1}{1}+\frac{1}{2}+\frac{1}{3}\ldots+\frac{1}{17}=\frac{h}{L_{17}}\] What is the remainder when $h$ is divided by $17$?

$\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

Solution 1

Notice that $L_{17}$ contains the highest power of every prime below $17$. Thus, $L_{17}=16\cdot 9 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \cdot 17$.

When writing the sum under a common fraction, we multiply the denominators by $L_{17}$ divided by each denominator. However, since $L_{17}$ is a multiple of $17$, all terms will be a multiple of $17$ until we divide out $17$, and the only term that will do this is $\frac{1}{17}$. Thus, the remainder of all other terms when divided by $17$ will be $0$, so the problem is essentially asking us what the remainder of $\frac{L_{17}}{17}$ divided by $17$ is. This is equivalent to finding the remainder of $16 \cdot 9 \cdot 5 \cdot 7 \cdot 11 \cdot 13$ divided by $17$.

We use modular arithmetic to simplify our answer:

This is congruent to $-1 \cdot 9 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \pmod{17}$.

Evaluating, we get: \begin{align*} (-1) \cdot 9 \cdot 35 \cdot 11 \cdot 13 &\equiv (-1) \cdot 9 \cdot 1 \cdot 11 \cdot 13 \pmod{17} \\ &\equiv 9 \cdot 11 \cdot (-13) \pmod{17} \\ &\equiv 9 \cdot 11 \cdot 4\pmod{17} \\ &\equiv 2 \cdot 11 \pmod{17} \\ &\equiv 5\pmod{17} \end{align*} Therefore the remainder is $\boxed{\textbf{(C) } 5}$.

~KingRavi

~mathboy282

~Scarletsyc

Solution 2

As in solution 1, we express the LHS as a sum under one common denominator. We note that \[\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{17} = \frac{\frac{17!}{1}}{17!} + \frac{\frac{17!}{2}}{17!} + \frac{\frac{17!}{3}}{17!} + \dots + \frac{\frac{17!}{17}}{17!}\]

Now, we have $h = L_{17}\left(\frac{\frac{17!}{1} + \frac{17!}{2} + \frac{17!}{3} + \dots + \frac{17!}{17}}{17!}\right)$. We'd like to find $h \pmod{17},$ so we can evaluate our expression $\pmod{17}.$ Since $\frac{\frac{17!}{1}}{17!}, \frac{\frac{17!}{2}}{17!}, \dots, \frac{\frac{17!}{16}}{17!}$ don't have a factor of $17$ in their denominators, and since $L_{17}$ is a multiple of $17,$ multiplying each of those terms and adding them will get a multiple of $17.$ $\pmod{17}$, that result is $0.$ Thus, we only need to consider $L_{17}\cdot \frac{\frac{17!}{17}}{17!} = \frac{L_{17}}{17} \pmod{17}.$ Proceed with solution $1$ to get $\boxed{\textbf{(C) }5}$.

~sirswagger21

Solution 3

Using Wolstenholmes' Theorem, we can rewrite $1 + \frac{1}{2} \dots + \frac{1}{16}$ as $\frac{17^2 n}{(17 - 1)!} = \frac{17^2 n}{16!}$ (for some $n \in \mathbb{Z}$). Adding the $\frac{1}{17}$ to $\frac{17^2 n}{16!}$, we get $\frac{17^3 n + 16!}{17!}$.

Now we have $\frac{17^3 n + 16!}{17!} = \frac{h}{L_{17}}$ and we want $h \pmod{17}$. We find that $\frac{L_{17}(17^3 n + 16!)}{17!} = \frac{L_{16}(17^3 n + 16!)}{16!} = h$. Taking $\pmod{17}$ and multiplying, we get $L_{16}(17^3 n + 16!) \equiv 16! \cdot h \pmod{17}$.

Applying Wilson's Theorem on $16!$ and reducing, we simplify the congruence to $L_{16}(0 - 1) \equiv -L_{16} \equiv -h \pmod{17}$. Now we proceed with Solution 1 and find that $L_{16} \equiv 5 \pmod{17}$, so our answer is $\boxed{\textbf{(C) }5}$.

~kn07

Video Solution By ThePuzzlr

https://youtu.be/TGcGamPXdNc

~ MathIsChess

See Also

2022 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 18 		Followed by
Problem 20
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=2022_AMC_10A_Problems/Problem_19&oldid=183843"