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 "2016 AMC 8 Problems/Problem 24"

(→‎Video Solution)
(→‎Solution 2)
Line 9: Line 9:
  
 
===Solution 2===  
 
===Solution 2===  
−
We know that out of <math>PQRST,</math> <math>QRS</math> is divisible by <math>5</math>. Therefore <math>S</math> is obviously 5 because <math>QRS</math> is divisible by 5. So we now have <math>PQR5T</math> as our number. Next, lets move on to the second piece of information that was given to us. RST is divisible by 3. So, according to the divisibility of 3 rule the sum of <math>RST</math> has to be a multiple of 3. The only 2 big enough is 9 and 12 and since 5 is already given. The possible sums of <math>RT</math> is 4 and 7. So, the possible values for <math>R</math> are 1,3,4,3 and the possible values of <math>T</math> is 3,1,3,4. So, using this we can move on to the fact that <math>PQR</math> is divisible by 4. So, using that we know that <math>R</math> has to be even so 4 is the only possible value for <math>R</math>. Using that we also know that 3 is the only possible value for 3. So, we know have <math>PQRST</math> = <math>PQ453</math> so the possible values are 1 and 2 for <math>P</math> and <math>Q</math>. Using the divisibility rule of 4 we know that <math>QR</math> has to be divisible by 4. So, either 14 or 24 are the possibilities, and 24 is divisible by 4. So the only value left for <math>P</math> is 1.  <math>P=\boxed{\textbf{(A)}\ 1}</math>.
+
We know that out of <math>PQRST,</math> <math>QRS</math> is divisible by <math>5</math>. Therefore <math>S</math> is obviously 5 because <math>QRS</math> is divisible by 5. So we now have <math>PQR5T</math> as our number. Next, let's move on to the second piece of information that was given to us. RST is divisible by 3. So, according to the divisibility by 3 rule, the sum of <math>RST</math> has to be a multiple of 3. The only 2 big enough are 9 and 12 and since 5 is already given. The possible sums of <math>RT</math> are 4 and 7. So, the possible values for <math>R</math> are 1,3,4,3 and the possible values of <math>T</math> are 3,1,3,4. So, using this we can move on to the fact that <math>PQR</math> is divisible by 4. So, using that we know that <math>R</math> has to be even so 4 is the only possible value for <math>R</math>. Using that we also know that 3 is the only possible value for 3. So, we have <math>PQRST</math> = <math>PQ453</math> so the possible values are 1 and 2 for <math>P</math> and <math>Q</math>. Using the divisibility rule of 4 we know that <math>QR</math> has to be divisible by 4. So, either 14 or 24 are the possibilities, and 24 is divisible by 4. So the only value left for <math>P</math> is 1.  <math>P=\boxed{\textbf{(A)}\ 1}</math>.
 +
 
 +
==Solution 3 (Divisibility Rules)==
 +
We know that <math>QRS</math> is divisible by <math>5</math>, so <math>S</math> would be either <math>5</math> or <math>0</math>. However, <math>0</math> is not a choice, so <math>S=5</math>. Also, <math>PQR</math> is divisble by <math>4</math>, so this means that <math>QR</math> is <math>12</math>, <math>32</math>, <math>24</math>, or <math>52</math>. If <math>R=2</math>, then <math>T</math> has to be <math>2</math> or <math>5</math> (<math>RST</math> is divisible by 3<math>), but both are taken. So, </math>R=4 \Rightarrow QR=24<math>. </math>R+S+T<math> must equal </math>9<math> or </math>12<math>, but because </math>4+5=9<math>, </math>R+S+T=12 \Rightarrow T=3<math>. This leaves </math>P=\boxed{\textbf{(A) }1}$
 +
 
 +
~MrThinker
  
 
==Video Solution==
 
==Video Solution==

Revision as of 09:43, 1 November 2022

Problem 24

The digits $1$, $2$, $3$, $4$, and $5$ are each used once to write a five-digit number $PQRST$. The three-digit number $PQR$ is divisible by $4$, the three-digit number $QRS$ is divisible by $5$, and the three-digit number $RST$ is divisible by $3$. What is $P$?

$\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

Solutions

Solution 1

We see that since $QRS$ is divisible by $5$, $S$ must equal either $0$ or $5$, but it cannot equal $0$, so $S=5$. We notice that since $PQR$ must be even, $R$ must be either $2$ or $4$. However, when $R=2$, we see that $T \equiv 2 \pmod{3}$, which cannot happen because $2$ and $5$ are already used up; so $R=4$. This gives $T \equiv 3 \pmod{4}$, meaning $T=3$. Now, we see that $Q$ could be either $1$ or $2$, but $14$ is not divisible by $4$, but $24$ is. This means that $Q=2$ and $P=\boxed{\textbf{(A)}\ 1}$.

Solution 2

We know that out of $PQRST,$ $QRS$ is divisible by $5$. Therefore $S$ is obviously 5 because $QRS$ is divisible by 5. So we now have $PQR5T$ as our number. Next, let's move on to the second piece of information that was given to us. RST is divisible by 3. So, according to the divisibility by 3 rule, the sum of $RST$ has to be a multiple of 3. The only 2 big enough are 9 and 12 and since 5 is already given. The possible sums of $RT$ are 4 and 7. So, the possible values for $R$ are 1,3,4,3 and the possible values of $T$ are 3,1,3,4. So, using this we can move on to the fact that $PQR$ is divisible by 4. So, using that we know that $R$ has to be even so 4 is the only possible value for $R$. Using that we also know that 3 is the only possible value for 3. So, we have $PQRST$ = $PQ453$ so the possible values are 1 and 2 for $P$ and $Q$. Using the divisibility rule of 4 we know that $QR$ has to be divisible by 4. So, either 14 or 24 are the possibilities, and 24 is divisible by 4. So the only value left for $P$ is 1. $P=\boxed{\textbf{(A)}\ 1}$.

Solution 3 (Divisibility Rules)

We know that $QRS$ is divisible by $5$, so $S$ would be either $5$ or $0$. However, $0$ is not a choice, so $S=5$. Also, $PQR$ is divisble by $4$, so this means that $QR$ is $12$, $32$, $24$, or $52$. If $R=2$, then $T$ has to be $2$ or $5$ ($RST$ is divisible by 3$), but both are taken. So,$R=4 \Rightarrow QR=24$.$R+S+T$must equal$9$or$12$, but because$4+5=9$,$R+S+T=12 \Rightarrow T=3$. This leaves$P=\boxed{\textbf{(A) }1}$

~MrThinker

Video Solution

https://youtu.be/WJ0Hodj0h2o - Happytwin

https://youtu.be/6xNkyDgIhEE?t=2905

https://youtu.be/4eKdjtugZUA

~savannahsolver

See Also

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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 AJHSME/AMC 8 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=2016_AMC_8_Problems/Problem_24&oldid=179918"