Difference between revisions of "2018 AMC 10B Problems/Problem 16"
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-TheMagician | -TheMagician | ||
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+ | ==Solution 4 (Lazy solution)== | ||
+ | Assume <math>a_1, a_2, ... a_2017</math> are multiples of 6 and find <math>2018^{2018} mod 6</math> (which happens to be 4). Then <math>{a_1}^3 + ... + {a_2018}^3</math> is congruent to <math>64 mod 6</math> or just <math>4</math>. | ||
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+ | Note: It is pretty easy to find this construction; one example is <math>a_1, a_2, ... a_{2016} = 6, a_{2018} = 4</math>, and then let <math>a_{2017}</math> be whatever is left that you need to add to get <math>2018^{2018}</math>. | ||
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+ | -Patrick4President | ||
==See Also== | ==See Also== |
Revision as of 14:04, 19 February 2018
Let be a strictly increasing sequence of positive integers such that What is the remainder when is divided by ?
Solution 1
By Euler's Totient Theorem, Alternatively, one could simply list out all the residues to the third power
Therefore the answer is congruent to
Solution 2
(not very good one)
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
We first note that . So what we are trying to find is what mod . We start by noting that is congruent to mod . So we are trying to find mod . 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 mod , thus giving our answer . You can prove this pattern using mods. But I thought this was easier.
-TheMagician
Solution 4 (Lazy solution)
Assume are multiples of 6 and find (which happens to be 4). Then is congruent to or just .
Note: It is pretty easy to find this construction; one example is , and then let be whatever is left that you need to add to get .
-Patrick4President
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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