Difference between revisions of "2021 AMC 12A Problems/Problem 9"
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Revision as of 02:54, 12 February 2021
Contents
Problem
Which of the following is equivalent to
![\[(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?\]](http://latex.artofproblemsolving.com/6/5/a/65a7155ec29794e20977d4bf1eef21987af9bbee.png)
Solution 1
All you need to do is multiply the entire equation by 
. Then all the terms will easily simplify by difference of squares and you will get 
or 
as your final answer. Notice you don't need to worry about 
because that's equal to 
.
-Lemonie
Solution 2
If you weren't able to come up with the 
insight, then you could just notice that the answer is divisible by
, and 
. We can then use Fermat's Little Theorem for 
on the answer choices to determine which of the answer choices are divisible by both 
and 
. This is .
-MEWTO
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution by OmegaLearn(Factorizations/Telescoping& Meta-solving)
~ pi_is_3.14
See also
| 2021 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 9 |
Followed by Problem 11 | |
| 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 | ||
| 2021 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 8 |
Followed by Problem 10 |
| 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 12 Problems and Solutions | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
