2021 AIME II Problems/Problem 9
Contents
Problem
Find the number of ordered pairs such that and are positive integers in the set and the greatest common divisor of and is not .
Solution 1
We make use of the (olympiad number theory) lemma that .
Noting , we have (by difference of squares) It is now easy to calculate the answer (with casework on ) as .
~Lcz
Solution 2 (Comprehensive and Generalized)
Claim (Solution 1's Lemma)
If and are positive integers such that then
There are two proofs to this claim, as shown below.
~MRENTHUSIASM
Proof 1 (Euclidean Algorithm)
If then from which the claim is clearly true.
Otherwise, let without the loss of generality. For all integers and such that the Euclidean Algorithm states that We apply this result repeatedly to reduce the larger number: Continuing, we have from which the proof is complete.
~MRENTHUSIASM
Proof 2 (Bézout's Identity)
Let It follows that and
By Bézout's Identity, there exist integers and such that so from which We know that
Next, we notice that Since is a common divisor of and we conclude that from which the proof is complete.
~MRENTHUSIASM
Solution (Detailed Explanation of Solution 1)
By the Euclidean Algorithm, we have We are given that Multiplying both sides by gives which implies that must have more factors of than does.
We construct the following table for the first positive integers: To count the ordered pairs we perform casework on the number of factors of that has:
- If has factors of then has options and has options. So, this case has ordered pairs.
- If has factor of then has options and has options. So, this case has ordered pairs.
- If has factors of then has options and has options. So, this case has ordered pairs.
- If has factors of then has options and has option. So, this case has ordered pairs.
Together, the answer is
~MRENTHUSIASM
Remark (GCD Property)
In of the solution, we use the following property of the greatest common divisor:
For positive integers and if then
As and are relatively prime (have no prime divisors in common), this property is intuitive.
~MRENTHUSIASM
See Also
2021 AIME II (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 | ||
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