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Difference between revisions of "2018 AMC 10B Problems/Problem 16"

(→‎Solution 6 (Nichomauss' Theorem))
(The proof was entirely incorrect. FLT proves the cases 1 and 5 mod 6, but none of the other cases. The last proof just showed in the cases 0, 1, and 5.)
(47 intermediate revisions by 22 users not shown)
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==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>
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Therefore, <cmath>a_1+a_2+\cdots+a_{2018} \equiv a_1^3+a_2^3+\cdots+a_{2018}^3 \pmod{6}.</cmath>
  
−
Note from Williamgolly: we can wlog assume <math>a_1,a_2... a_2017=0(mod6)</math> and have <math>a_2018=2(mod6)</math> to make life easier
+
Thus the answer is congruent to <math>2018^{2018}\equiv 2^{2018} \pmod{6} = \boxed{ \text{(E)}4}</math> because <math>2^n \pmod{6}</math> alternates with <math>2</math> and <math>4</math> when <math>n</math> increases.
 +
 
 +
~Dolphindesigner
 +
 
 +
~Major error correction made by akashsuresh1.22~
  
 
==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>
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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 (Partial Proof)==
 
==Solution 3 (Partial Proof)==
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==Solution 4 (Lazy solution)==
 
==Solution 4 (Lazy solution)==
−
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>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 6 (Nichomauss' 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>.
 +
 
 +
==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
  
−
Seeing the cubes of numbers, we think of Nichomauss's theorem, which states that <math>(a_1^3 + a_2^3 + ... + a_n^3) = (a_1 + a_2 + ... + a_n)^2</math>. We can do this and deduce that <math>(a_1^3 + a_2^3 + ... + a_{2018}^3) = 2018^{2018}</math> squared.
+
~bunny1
  
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Now, we find <math>2018\mod 6</math>, which is 2. This means that we need to find <math>2^{2018} \mod {6}</math>, which we can find using a pattern to be <math>4</math>. Therefore, the answer is <math>4^3\mod 6</math>, which is congruent to <math>\boxed{\textbf{(E)} 4}</math>
+
== Video Solution 3 by OmegaLearn==
 +
https://youtu.be/4_x1sgcQCp4?t=112
  
−
-ericshi1685
+
~ pi_is_3.14
  
 
==See Also==
 
==See Also==

Revision as of 18:27, 5 March 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, \[a_1+a_2+\cdots+a_{2018} \equiv a_1^3+a_2^3+\cdots+a_{2018}^3 \pmod{6}.\]

Thus the answer is congruent to $2018^{2018}\equiv 2^{2018} \pmod{6} = \boxed{ \text{(E)}4}$ because $2^n \pmod{6}$ alternates with $2$ and $4$ when $n$ increases.

~Dolphindesigner

~Major error correction made by akashsuresh1.22~

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 $a$.

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

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