Difference between revisions of "2018 AMC 10B Problems/Problem 16"
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==Solution 3 (Partial Proof)== | ==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 | + | 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+...+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. | 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. | ||
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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>. | |
-Patrick4President | -Patrick4President |
Revision as of 16:30, 27 January 2020
Contents
Problem
Let be a strictly increasing sequence of positive integers such that What is the remainder when is divided by ?
Solution 1
One could simply list out all the residues to the third power . (Edit: Euler's totient theorem is not a valid approach to showing that they are all congruent . This is due to the fact that need not be relatively prime to .)
Therefore the answer is congruent to
Note from Williamgolly: we can wlog assume and have to make life easier
Solution 2
Note that
Note that Therefore, .
Thus, . 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
Solution 3 (Partial Proof)
First, we can assume that the problem will have a consistent answer for all possible values of . For the purpose of this solution, we will assume that .
We first note that . So what we are trying to find is what mod . We start by noting that is congruent to . So we are trying to find . Instead of trying to do this with some number theory skills, we could just look for a pattern. We start with small powers of and see that is mod , is mod , is mod , is mod , and so on... So we see that since has an even power, it must be congruent to , thus giving our answer . 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 . For the purpose of this solution, assume are multiples of 6 and find (which happens to be ). Then is congruent to or just .
-Patrick4President
Solution 5 (Fermat's Little Theorem)
First note that each by Fermat's Little Theorem. This implies that . Also, all , hence by Fermat's Little Theorem.Thus, . Now set . Then, we have the congruences and . By the Chinese Remainder Theorem, a solution must exist, and indeed solving the congruence we get that . Thus, the answer is
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.