Difference between revisions of "2021 AMC 10A Problems/Problem 11"
MRENTHUSIASM (talk | contribs) m (→Solution 1) |
(→Solution 3 (Educated Guess): formalize) |
||
Line 25: | Line 25: | ||
- icecreamrolls8 | - icecreamrolls8 | ||
− | ==Solution 3 ( | + | ==Solution 3 (Residues)== |
− | + | By definition of bases, this number is a polynomial in terms of <math>b</math> (<math>(2b^3+0b^2+2b^1+1b^0)-(2b^2+2b^1+1b^0)</math> to be exact). Thus, for two values of <math>b</math> that share the same residue modulo <math>3</math>, the two resulting numbers will share the same residue modulo <math>3</math>, so either both or neither will be divisible by <math>3</math>. | |
− | ~MRENTHUSIASM | + | Choices <math>\textbf{(A)} ~3,\textbf{(B)} ~4,\textbf{(C)} ~6,\textbf{(D)} ~7,\textbf{(E)} ~8</math> are congruent to <math>0,1,0,1,2</math> modulo <math>3</math>, respectively. This means <math>\textbf{(A)} ~3</math> and <math>\textbf{(C)} ~6</math> are either both wrong or both right (and the latter obviously cannot be the case), and likewise with <math>\textbf{(B)} ~4</math> and <math>\textbf{(D)} ~7</math>. This leaves <math>\boxed{\textbf{(E)} ~8}</math>, the only choice with a unique residue. |
+ | |||
+ | ~MRENTHUSIASM (revised by [[User:emerald_block|emerald_block]]) | ||
==Video Solution (Simple and Quick)== | ==Video Solution (Simple and Quick)== |
Revision as of 12:03, 8 November 2021
Contents
[hide]Problem
For which of the following integers is the base- number not divisible by ?
Solution 1 (Factor)
We have which is divisible by unless The only choice congruent to modulo is
~MRENTHUSIASM
Solution 2 (Easy)
Vertically subtracting we see that the ones place becomes 0, and so does the place. Then, we perform a carry (make sure the carry is in base !). Let . Then, we have our final number as
Now, when expanding, we see that this number is simply .
Now, notice that the final number will only be congruent to if either , or if (because note that would become , and would become as well, and therefore the final expression would become . Therefore, must be . Among the answers, only 8 is , and therefore our answer is
- icecreamrolls8
Solution 3 (Residues)
By definition of bases, this number is a polynomial in terms of ( to be exact). Thus, for two values of that share the same residue modulo , the two resulting numbers will share the same residue modulo , so either both or neither will be divisible by .
Choices are congruent to modulo , respectively. This means and are either both wrong or both right (and the latter obviously cannot be the case), and likewise with and . This leaves , the only choice with a unique residue.
~MRENTHUSIASM (revised by emerald_block)
Video Solution (Simple and Quick)
~ Education, the Study of Everything
Video Solution
https://www.youtube.com/watch?v=XBfRVYx64dA&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=10
~North America Math Contest Go Go Go
Video Solution 3
~savannahsolver
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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.