Art of Problem Solving
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

2016 AMC 8 Problems/Problem 5: Difference between revisions

← Older edit
Ab2024 (talk | contribs)
editor
64 edits
Timi821 (talk | contribs)
editor
9 edits
 
(25 intermediate revisions by 15 users not shown)
Line 1: Line 1:
== Problem ==
The number <math>N</math> is a two-digit number.
The number <math>N</math> is a two-digit number.


Line 10: Line 12:
<math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7</math>
<math>\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7</math>


==Video Solution==
==Solution 1==
https://youtu.be/7an5wU9Q5hk?t=574


==Solution 1==


From the second bullet point, we know that the second digit must be <math>3</math>. Because there is a remainder of <math>1</math> when it is divided by <math>9</math>, the multiple of <math>9</math> must end in a <math>2</math> in order for it to have the desired remainder<math>\pmod {10}.</math> We now look for this one:  
From the second bullet point, we know that the second digit must be <math>3</math>, for a number divisible by <math>10</math> ends in zero. Since there is a remainder of <math>1</math> when <math>N</math> is divided by <math>9</math>, the multiple of <math>9</math> must end in a <math>2</math> for it to have the desired remainder<math>\pmod {10}.</math> We now look for this one:  


<math>9(1)=9\\
<math>9(1)=9\\
Line 28: Line 28:
The number <math>72+1=73</math> satisfies both conditions. We subtract the biggest multiple of <math>11</math> less than <math>73</math> to get the remainder. Thus, <math>73-11(6)=73-66=\boxed{\textbf{(E) }7}</math>.
The number <math>72+1=73</math> satisfies both conditions. We subtract the biggest multiple of <math>11</math> less than <math>73</math> to get the remainder. Thus, <math>73-11(6)=73-66=\boxed{\textbf{(E) }7}</math>.


==Solution 2 ~ More efficient for proofs==
~CHECKMATE2021
 
==Solution 2==
We know that the number has to be one more than a multiple of <math>9</math>, because of the remainder of one, and the number has to be <math>3</math> more than a multiple of <math>10</math>, which means that it has to end in a <math>3</math>. Now, if we just list the first few multiples of <math>9</math> adding one to the number we get: <math>10, 19, 28, 37, 46, 55, 64, 73, 82, 91</math>. As we can see from these numbers,  the only one that has a three in the units place is <math>73</math>, thus we divide <math>73</math> by <math>11</math>, getting <math>6</math> <math>R7</math>, hence, <math>\boxed{\textbf{(E) }7}</math>.
-fn106068


This two digit number must take the form of <math>10x+y,</math> where <math>x</math> and <math>y</math> are integers <math>0</math> to <math>9.</math> However, if x is an integer, we must have <math>y=3.</math> So, the number's new form is <math>10x+3.</math> This needs to have a remainder of <math>1</math> when divided by <math>9.</math> Because of the <math>9</math> divisibility rule, we have <cmath>10x+3 \equiv 1 \pmod 9.</cmath>
We could also remember that, for a two-digit number to be divisible by <math>9</math>, the sum of its digits has to be equal to <math>9</math>. Since the number is one more than a multiple of <math>9</math>, the multiple we are looking for has a ones digit of <math>2</math>, and therefore a tens digit of <math>9-2 = 7</math>, and then we could proceed as above. -vaisri
We subtract the three, getting <cmath>10x \equiv -2 \pmod 9.</cmath>
which simplifies to <cmath>10x \equiv 7 \pmod 9.</cmath>
However, <math>9x \equiv 0 \pmod 9,</math> so <cmath>10x - 9x \equiv 7 - 0 \pmod 9</cmath> and <cmath>x \equiv 7 \pmod 9.</cmath>


Let the quotient of <math>9</math> in our modular equation be <math>c,</math> and let our desired number be <math>z,</math> so <math>x=9c+7</math> and <math>z = 10x+3.</math>  We substitute these values into <math>z = 10x+3,</math> and get <cmath> z = 10(9c+7) + 3</cmath> so <cmath> z = 90c+73.</cmath> As a result, <math>z \equiv 73 \pmod {90}.</math>
==Video Solution==


*Alternatively, we could have also used a system of modular equations to immediately receive <math>z \equiv 73 \pmod {90}.</math>
https://youtu.be/d-bCEDoZEjg?si=VFLhpgyJ_vHhE7h3


To prove generalization vigorously, we can let <math>a</math> be the remainder when <math>z</math> is divided by <math>11.</math> Setting up a modular equation, we have <cmath>90c + 73 \equiv a \pmod {11}.</cmath> Simplifying, <cmath>90c+7 \equiv a \pmod {11}</cmath> If <math>c = 1,</math> then we don't have a 2 digit number! Thus, <math>c=0</math> and <math>a=\boxed { \textbf{(E) }7}</math>
A solution so simple a 12-year-old made it!


==Video Solution==
~Elijahman~
 
==Video Solution by OmegaLearn==
https://youtu.be/7an5wU9Q5hk?t=574
 
==Solution 3==
 
We can set up a system of modular congruences
n = 1 (mod 9)
n = 3 (mod 10)
 
9x + 1 = 3 (mod 10)
    -1  -1
 
9x = 2(mod 10)
    +70
 
9x = 72(mod 10)
/9  /9
 
x = 8 (mod 10)
 
10f + 8
9(10f+8)+1= n
90f + 72 + 1 = n
90f + 73  = n
n = 73 (mod 90)
 
73/11 = 6 r7


https://www.youtube.com/watch?v=LqnQQcUVJmA (has questions 1-5)
- timi821


==Video Solution==
https://youtu.be/aKWQl7kEMy0


{{AMC8 box|year=2016|num-b=4|num-a=6}}
~savannahsolver
{{MAA Notice}}

Latest revision as of 09:32, 9 November 2025

Problem

The number $N$ is a two-digit number.

• When $N$ is divided by $9$, the remainder is $1$.

• When $N$ is divided by $10$, the remainder is $3$.

What is the remainder when $N$ is divided by $11$?


$\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad \textbf{(E) }7$

Solution 1

From the second bullet point, we know that the second digit must be $3$, for a number divisible by $10$ ends in zero. Since there is a remainder of $1$ when $N$ is divided by $9$, the multiple of $9$ must end in a $2$ for it to have the desired remainder$\pmod {10}.$ We now look for this one:

$9(1)=9\\ 9(2)=18\\ 9(3)=27\\ 9(4)=36\\ 9(5)=45\\ 9(6)=54\\ 9(7)=63\\ 9(8)=72$

The number $72+1=73$ satisfies both conditions. We subtract the biggest multiple of $11$ less than $73$ to get the remainder. Thus, $73-11(6)=73-66=\boxed{\textbf{(E) }7}$.

~CHECKMATE2021

Solution 2

We know that the number has to be one more than a multiple of $9$, because of the remainder of one, and the number has to be $3$ more than a multiple of $10$, which means that it has to end in a $3$. Now, if we just list the first few multiples of $9$ adding one to the number we get: $10, 19, 28, 37, 46, 55, 64, 73, 82, 91$. As we can see from these numbers, the only one that has a three in the units place is $73$, thus we divide $73$ by $11$, getting $6$ $R7$, hence, $\boxed{\textbf{(E) }7}$. -fn106068

We could also remember that, for a two-digit number to be divisible by $9$, the sum of its digits has to be equal to $9$. Since the number is one more than a multiple of $9$, the multiple we are looking for has a ones digit of $2$, and therefore a tens digit of $9-2 = 7$, and then we could proceed as above. -vaisri

Video Solution

https://youtu.be/d-bCEDoZEjg?si=VFLhpgyJ_vHhE7h3

A solution so simple a 12-year-old made it!

~Elijahman~

Video Solution by OmegaLearn

https://youtu.be/7an5wU9Q5hk?t=574

Solution 3

We can set up a system of modular congruences 

n = 1 (mod 9)
n = 3 (mod 10)

9x + 1 = 3 (mod 10) 

   -1   -1

9x = 2(mod 10) 

    +70

9x = 72(mod 10) /9 /9

x = 8 (mod 10)

10f + 8 9(10f+8)+1= n 90f + 72 + 1 = n 90f + 73 = n n = 73 (mod 90)

73/11 = 6 r7

- timi821

Video Solution

https://youtu.be/aKWQl7kEMy0

~savannahsolver

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=2016_AMC_8_Problems/Problem_5&oldid=267971"