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 (Vertical Subtraction)
Vertically subtracting we see that the ones place becomes
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
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.