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 "2018 AMC 10B Problems/Problem 16"

m (→‎Solution 3)
(→‎Solution 1)
 
(61 intermediate revisions by 29 users not shown)
Line 8: Line 8:
 
==Solution 1==
 
==Solution 1==
  
−
One could simply list out all the residues to the third power <math>\mod 6</math>. (Edit: Euler's totient theorem is not a valid approach to showing that they are all congruent <math>\mod 6</math>. This is due to the fact that <math>a_k</math> need not be relatively prime to <math>6</math>.)
+
Verify that <math>a^3 \equiv a \pmod{6}</math> manually for all <math>a\in \mathbb{Z}/6\mathbb{Z}</math>. We check: <math>0^3 \equiv 0 \pmod{6}</math>, <math>1^3 \equiv 1 \pmod{6}</math>, <math>2^3 \equiv 8 \equiv 2 \pmod{6}</math>, <math>3^3 \equiv 27 \equiv 3 \pmod{6}</math>, <math>4^3 \equiv 64 \equiv 4 \pmod{6}</math>, and <math>5^3 \equiv 125 \equiv 5 \pmod{6}</math>. We conclude that <math>a^3 \equiv a \pmod{6}</math>.
  
−
Therefore the answer is congruent to <math>2018^{2018}\equiv 2^{2018} \pmod{6} = \boxed{ (E)4}</math>
+
Therefore, <cmath>a_1^3+a_2^3+\cdots+a_{2018}^3 \equiv a_1+a_2+\cdots+a_{2018}\pmod{6}.</cmath>
 +
 
 +
Thus the answer is congruent to <math>2018^{2018} = 1009^{2018} \cdot 2^{2018} \equiv \boxed{ \text{(E)}4} \pmod{6}</math> because <math>2^n \pmod{6}</math> alternates with <math>2</math> and <math>4</math> when <math>n</math> increases, and by [[Euler's Totient Theorem]] <math>1009^{2018} = (1009^2)^{1009} \equiv 1^{1009} \equiv 1 \pmod{6}</math>.
 +
 
 +
~Dolphindesigner
 +
 
 +
~Major error correction made by akashsuresh1.22~
 +
 
 +
~Another major error correction made by YTH
 +
 
 +
~Glossed-over detail filled in by GAMER100
  
 
==Solution 2==
 
==Solution 2==
  
−
Note that <math>\left(a_1+a_2+\cdots+a_{2018}\right)^3=a_1^3+a_2^3+\cdots+a_{2018}^3+3a_1^2\left(a_1+a_2+\cdots+a_{2018}-a_1\right)+3a_2^2\left(a_1+a_2+\cdots+a_{2018}-a_2\right)+\cdots+3a_{2018}^2\left(a_1+a_2+\cdots+a_{2018}-a_{2018}\right)+6\sum_{i\neq j\neq k}^{2018} a_ia_ja_k</math>
+
Note that <math>\left(a_1+a_2+\cdots+a_{2018}\right)^3=a_1^3+a_2^3+\cdots+a_{2018}^3+3a_1^2\left(a_1+a_2\\
 +
+\cdots+a_{2018}-a_1\right)+3a_2^2\left(a_1+a_2+\cdots+a_{2018}-a_2\right)+\cdots+3a_{2018}^2\left(a_1+a_2+\cdots+a_{2018}-a_{2018}\right)+6\sum_{i\neq j\neq k}^{2018} a_ia_ja_k</math>
  
 
Note that <math>
 
Note that <math>
Line 22: Line 33:
 
Therefore, <math>-2(a_1^3+a_2^3+\cdots+a_{2018}^3)\equiv \left(2018^{2018}\right)^3\equiv\left( 2^{2018}\right)^3\equiv 4^3\equiv 4\pmod{6}</math>.
 
Therefore, <math>-2(a_1^3+a_2^3+\cdots+a_{2018}^3)\equiv \left(2018^{2018}\right)^3\equiv\left( 2^{2018}\right)^3\equiv 4^3\equiv 4\pmod{6}</math>.
  
−
Thus, <math>a_1^3+a_2^3+\cdots+a_{2018}^3\equiv 1\pmod 3</math>. However, since cubing preserves parity, and the sum of the individual terms is even, the some of the cubes is also even, and our answer is <math>\boxed{\text{(E) }4}</math>
+
Thus, <math>a_1^3+a_2^3+\cdots+a_{2018}^3\equiv 1\pmod 3</math>. However, since cubing preserves parity, and the sum of the individual terms is even, the sum of the cubes is also even, and our answer is <math>\boxed{\text{(E) }4}</math>
  
−
==Solution 3==
+
==Solution 3 (Partial Proof)==
 +
First, we can assume that the problem will have a consistent answer for all possible values of <math>a_1</math>. For the purpose of this solution, we will assume that <math>a_1 = 1</math>.
  
−
We first note that <math>1^3+2^3+...=(1+2+...)^2</math>. So what we are trying to find is what <math>\left(2018^{2018}\right)^2=\left(2018^{4036}\right)</math> mod <math>6</math>. We start by noting that <math>2018</math> is congruent to <math>2 \pmod{6}</math>. So we are trying to find <math>\left(2^{4036}\right) \pmod{6}</math>. Instead of trying to do this with some number theory skills, we could just look for a pattern. We start with small powers of <math>2</math> and see that <math>2^1</math> is <math>2</math> mod <math>6</math>, <math>2^2</math> is <math>4</math> mod <math>6</math>, <math>2^3</math> is <math>2</math> mod <math>6</math>, <math>2^4</math> is <math>4</math> mod <math>6</math>, and so on... So we see that since <math>\left(2^{4036}\right)</math> has an even power, it must be congruent to <math>4 \pmod{6}</math>, thus giving our answer <math>\boxed{\text{(E) }4}</math>. You can prove this pattern using mods. But I thought this was easier.
+
We first note that <math>1^3+2^3+...+n^3 = (1+2+...+n)^2</math>. So what we are trying to find is what <math>\left(2018^{2018}\right)^2=\left(2018^{4036}\right)</math> mod <math>6</math>. We start by noting that <math>2018</math> is congruent to <math>2 \pmod{6}</math>. So we are trying to find <math>\left(2^{4036}\right) \pmod{6}</math>. Instead of trying to do this with some number theory skills, we could just look for a pattern. We start with small powers of <math>2</math> and see that <math>2^1</math> is <math>2</math> mod <math>6</math>, <math>2^2</math> is <math>4</math> mod <math>6</math>, <math>2^3</math> is <math>2</math> mod <math>6</math>, <math>2^4</math> is <math>4</math> mod <math>6</math>, and so on... So we see that since <math>\left(2^{4036}\right)</math> has an even power, it must be congruent to <math>4 \pmod{6}</math>, thus giving our answer <math>\boxed{\text{(E) }4}</math>. You can prove this pattern using mods. But I thought this was easier.
  
 
-TheMagician
 
-TheMagician
  
 
==Solution 4 (Lazy solution)==
 
==Solution 4 (Lazy solution)==
−
Assume <math>a_1, a_2, ... a_{2017}</math> are multiples of 6 and find <math>2018^{2018} \pmod{6}</math> (which happens to be 4). Then <math>{a_1}^3 + ... + {a_{2018}}^3</math> is congruent to <math>64 \pmod{6}</math> or just <math>4</math>.  
+
First, we can assume that the problem will have a consistent answer for all possible values of <math>a_1</math>. For the purpose of this solution, assume <math>a_1, a_2, ... a_{2017}</math> are multiples of 6 and find <math>2018^{2018} \pmod{6}</math> (which happens to be <math>4</math>). Then <math>{a_1}^3 + ... + {a_{2018}}^3</math> is congruent to <math>64 \pmod{6}</math> or just <math>\boxed{\textbf{(E)}  4}</math>.  
  
 
-Patrick4President
 
-Patrick4President
  
−
==Solution 5 (Fermat's Little Theorem)==
+
~minor edit made by CatachuKetchup~
 +
 
 +
==Solution 5 (Even Lazier Solution)==
 +
Due to the large amounts of variables in the problem, and the fact that the test is only 75 minutes, you can assume that the answer is probably just <math>2018^{2018} \pmod{6} </math>, which is <math>\boxed{\textbf{(E)}  4}</math>.
 +
 
 +
~ Zeeshan12 [Be warned that this technique is not recommended for all problems and you should use it as a last resort]
 +
 
 +
==Algebraic Insight into Given Property==
 +
Mods is a good way to prove <math>a^3 \equiv a \pmod6</math>: residues are simply <math>3, \pm 2, \pm 1, 0</math>. Only <math>2</math> and <math>3</math> are necessary to check.
 +
Another way is to observe that <math>a^3-a</math> factors into <math>(a-1)a(a+1)</math>. Any <math>k</math> consecutive numbers must be a multiple of <math>k</math>, so <math>a^3-a</math> is both divisible by <math>2</math> and <math>3</math>. This provides an algebraic method for proving <math>a^3 \equiv a \pmod6</math> for all <math>a</math>.
 +
 
 +
==Solution 6==
 +
We are not given any other requirements than the two given in the problem, so we can assume that <math>a_1 + a_{2018} = a_2 + a_{2017} + \cdots + a_{1004} + a_{1005} = \frac{2018^{2018}}{2018} \cdot 2</math>. Since the (theoretical) <math>a_{1004.5}</math> averages out to be <math>2018^{2017}</math> (lets call this constant <math>N</math>), we can let <math>(a_1, a_{2018}) = (N - x_1, N+x_1)</math>, <math>(a_2, a_{2017}) = (N - x_2, N+x_2)</math>, ... ,<math>(a_{1004}, a_{1005}) = (N - x_{1004}, N+x_{1004})</math> for the sequence of decreasing positive integers <math>x_1, x_2, \cdots, x_{1004}</math>. Notice that for any such pair (say <math>(a_{n}, a_{2019-n}</math>), we have <math>(a_n)^3 + (a_{2019-n})^3 = (N - x_n)^3 + (N+x_n)^3 = 2N^3 + 6N\cdot x_n \equiv 2N^3 mod 6</math>. Thus, <math>a_1^3 + a_2^3 + \cdots + a_{2018}^3 \equiv 1004\cdot 2N^3 \equiv 2018(2018^{2017})^3 \equiv 2018^{6052} \equiv 2^{6052} mod 6</math>. Since <math>2^x mod 6</math> repeats in the pattern <math>2, 4, 2, 4 \cdots</math>, our final answer is <math>\boxed{\textbf{(E)}  4}</math>
 +
 
 +
~ Lucas.xue
 +
 
 +
==Video Solution 1==
 +
With Modular Arithmetic Intro
 +
https://www.youtube.com/watch?v=wbv3TArroSs
 +
 
 +
~IceMatrix
 +
 
 +
==Video Solution 2==
 +
https://www.youtube.com/watch?v=SRjZ6B5DR74
 +
 
 +
~bunny1
 +
 
 +
== Video Solution 3 by OmegaLearn==
 +
https://youtu.be/4_x1sgcQCp4?t=112
 +
 
 +
~ pi_is_3.14
  
−
First note that each <math>a_{i}^3 \equiv a_{i} \pmod 3</math> by Fermat's Little Theorem. This implies that <math>a_{1}^3+...+a_{2018}^3 \equiv a_{1}+...+a_{2018} \equiv 2^{2018} \equiv 1 \pmod{3}</math>. Also, all <math>a_{i}^2 \equiv a_{i} \pmod{2}</math>, hence <math>a_{i}^3 \equiv (a_{i})(a_{i}^2) \equiv a_{i}^2 \equiv a_{i} \pmod{2}</math> by Fermat's Little Theorem.Thus, <math>a_{1}^3+...a_{2018}^3 \equiv 2^{2018} \equiv 0 \pmod{2}</math>. Now set <math>x=a_{1}^3+...+a_{2018}^3</math>. Then, we have the congruences <math>x \equiv 0 \pmod{2}</math> and <math>x \equiv 1 \pmod{3}</math>. By the Chinese Remainder Theorem, a solution must exist, and indeed solving the congruence we get that <math>x \equiv 4 \pmod{6}</math>. Thus, the answer is <math>\boxed{ (E)4}</math>
 
 
==See Also==
 
==See Also==
  
 
{{AMC10 box|year=2018|ab=B|num-b=15|num-a=17}}
 
{{AMC10 box|year=2018|ab=B|num-b=15|num-a=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:50, 2 November 2024

Problem

Let $a_1,a_2,\dots,a_{2018}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2018}=2018^{2018}.\] What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$?

$\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4$

Solution 1

Verify that $a^3 \equiv a \pmod{6}$ manually for all $a\in \mathbb{Z}/6\mathbb{Z}$. We check: $0^3 \equiv 0 \pmod{6}$, $1^3 \equiv 1 \pmod{6}$, $2^3 \equiv 8 \equiv 2 \pmod{6}$, $3^3 \equiv 27 \equiv 3 \pmod{6}$, $4^3 \equiv 64 \equiv 4 \pmod{6}$, and $5^3 \equiv 125 \equiv 5 \pmod{6}$. We conclude that $a^3 \equiv a \pmod{6}$.

Therefore, \[a_1^3+a_2^3+\cdots+a_{2018}^3 \equiv a_1+a_2+\cdots+a_{2018}\pmod{6}.\]

Thus the answer is congruent to $2018^{2018} = 1009^{2018} \cdot 2^{2018} \equiv \boxed{ \text{(E)}4} \pmod{6}$ because $2^n \pmod{6}$ alternates with $2$ and $4$ when $n$ increases, and by Euler's Totient Theorem $1009^{2018} = (1009^2)^{1009} \equiv 1^{1009} \equiv 1 \pmod{6}$.

~Dolphindesigner

~Major error correction made by akashsuresh1.22~

~Another major error correction made by YTH

~Glossed-over detail filled in by GAMER100

Solution 2

Note that $\left(a_1+a_2+\cdots+a_{2018}\right)^3=a_1^3+a_2^3+\cdots+a_{2018}^3+3a_1^2\left(a_1+a_2\\ +\cdots+a_{2018}-a_1\right)+3a_2^2\left(a_1+a_2+\cdots+a_{2018}-a_2\right)+\cdots+3a_{2018}^2\left(a_1+a_2+\cdots+a_{2018}-a_{2018}\right)+6\sum_{i\neq j\neq k}^{2018} a_ia_ja_k$

Note that $a_1^3+a_2^3+\cdots+a_{2018}^3+3a_1^2\left(a_1+a_2+\cdots+a_{2018}-a_1\right)+3a_2^2\left(a_1+a_2+\cdots+a_{2018}-a_2\right)+\cdots+3a_{2018}^2\left(a_1+a_2+\cdots+a_{2018}-a_{2018}\right)+6\sum_{i\neq j\neq k}^{2018} a_ia_ja_k\equiv a_1^3+a_2^3+\cdots+a_{2018}^3+3a_1^2({2018}^{2018}-a_1)+3a_2^2({2018}^{2018}-a_2)+\cdots+3a_{2018}^2({2018}^{2018}-a_{2018}) \equiv -2(a_1^3+a_2^3+\cdots+a_{2018}^3)\pmod 6$ Therefore, $-2(a_1^3+a_2^3+\cdots+a_{2018}^3)\equiv \left(2018^{2018}\right)^3\equiv\left( 2^{2018}\right)^3\equiv 4^3\equiv 4\pmod{6}$.

Thus, $a_1^3+a_2^3+\cdots+a_{2018}^3\equiv 1\pmod 3$. However, since cubing preserves parity, and the sum of the individual terms is even, the sum of the cubes is also even, and our answer is $\boxed{\text{(E) }4}$

Solution 3 (Partial Proof)

First, we can assume that the problem will have a consistent answer for all possible values of $a_1$. For the purpose of this solution, we will assume that $a_1 = 1$.

We first note that $1^3+2^3+...+n^3 = (1+2+...+n)^2$. So what we are trying to find is what $\left(2018^{2018}\right)^2=\left(2018^{4036}\right)$ mod $6$. We start by noting that $2018$ is congruent to $2 \pmod{6}$. So we are trying to find $\left(2^{4036}\right) \pmod{6}$. Instead of trying to do this with some number theory skills, we could just look for a pattern. We start with small powers of $2$ and see that $2^1$ is $2$ mod $6$, $2^2$ is $4$ mod $6$, $2^3$ is $2$ mod $6$, $2^4$ is $4$ mod $6$, and so on... So we see that since $\left(2^{4036}\right)$ has an even power, it must be congruent to $4 \pmod{6}$, thus giving our answer $\boxed{\text{(E) }4}$. You can prove this pattern using mods. But I thought this was easier.

-TheMagician

Solution 4 (Lazy solution)

First, we can assume that the problem will have a consistent answer for all possible values of $a_1$. For the purpose of this solution, assume $a_1, a_2, ... a_{2017}$ are multiples of 6 and find $2018^{2018} \pmod{6}$ (which happens to be $4$). Then ${a_1}^3 + ... + {a_{2018}}^3$ is congruent to $64 \pmod{6}$ or just $\boxed{\textbf{(E)} 4}$.

-Patrick4President

~minor edit made by CatachuKetchup~

Solution 5 (Even Lazier Solution)

Due to the large amounts of variables in the problem, and the fact that the test is only 75 minutes, you can assume that the answer is probably just $2018^{2018} \pmod{6}$, which is $\boxed{\textbf{(E)} 4}$.

~ Zeeshan12 [Be warned that this technique is not recommended for all problems and you should use it as a last resort]

Algebraic Insight into Given Property

Mods is a good way to prove $a^3 \equiv a \pmod6$: residues are simply $3, \pm 2, \pm 1, 0$. Only $2$ and $3$ are necessary to check. Another way is to observe that $a^3-a$ factors into $(a-1)a(a+1)$. Any $k$ consecutive numbers must be a multiple of $k$, so $a^3-a$ is both divisible by $2$ and $3$. This provides an algebraic method for proving $a^3 \equiv a \pmod6$ for all $a$.

Solution 6

We are not given any other requirements than the two given in the problem, so we can assume that $a_1 + a_{2018} = a_2 + a_{2017} + \cdots + a_{1004} + a_{1005} = \frac{2018^{2018}}{2018} \cdot 2$. Since the (theoretical) $a_{1004.5}$ averages out to be $2018^{2017}$ (lets call this constant $N$), we can let $(a_1, a_{2018}) = (N - x_1, N+x_1)$, $(a_2, a_{2017}) = (N - x_2, N+x_2)$, ... ,$(a_{1004}, a_{1005}) = (N - x_{1004}, N+x_{1004})$ for the sequence of decreasing positive integers $x_1, x_2, \cdots, x_{1004}$. Notice that for any such pair (say $(a_{n}, a_{2019-n}$), we have $(a_n)^3 + (a_{2019-n})^3 = (N - x_n)^3 + (N+x_n)^3 = 2N^3 + 6N\cdot x_n \equiv 2N^3 mod 6$. Thus, $a_1^3 + a_2^3 + \cdots + a_{2018}^3 \equiv 1004\cdot 2N^3 \equiv 2018(2018^{2017})^3 \equiv 2018^{6052} \equiv 2^{6052} mod 6$. Since $2^x mod 6$ repeats in the pattern $2, 4, 2, 4 \cdots$, our final answer is $\boxed{\textbf{(E)} 4}$

~ Lucas.xue

Video Solution 1

With Modular Arithmetic Intro https://www.youtube.com/watch?v=wbv3TArroSs

~IceMatrix

Video Solution 2

https://www.youtube.com/watch?v=SRjZ6B5DR74

~bunny1

Video Solution 3 by OmegaLearn

https://youtu.be/4_x1sgcQCp4?t=112

~ pi_is_3.14

See Also

2018 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 15 		Followed by
Problem 17
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=2018_AMC_10B_Problems/Problem_16&oldid=231275"