2020 AMC 10B Problems/Problem 22
Contents
Problem
What is the remainder when is divided by
?
Solution 1
Let . We are now looking for the remainder of
.
We could proceed with polynomial division, but the denominator looks awfully similar to the Sophie Germain Identity, which states that
Let's use the identity, with and
, so we have
Rearranging, we can see that this is exactly what we need:
So
Since the first half divides cleanly as shown earlier, the remainder must be ~quacker88
Solution 2
Similar to Solution 1, let . It suffices to find remainder of
. Dividing polynomials results in a remainder of
.
Solution 3 (MAA Original Solution)
Thus, we see that the remainder is surely
(Source: https://artofproblemsolving.com/community/c5h2001950p14000817)
Solution 4
We let and
.
Next we write
.
We know that
by the Sophie Germain identity so to find
we find that
which shows that the remainder is
Solution 5
We let . That means
and
. Then, we simply do polynomial division, and find that the remainder is
.
Video Solution 1 by Beauty Of Math
Video Solution 2 by the Beauty Of Math
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
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