2021 AMC 10A Problems/Problem 11
Contents
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 | |
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All AMC 10 Problems and Solutions |
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