Math Problem Statement
if prime p and q such that (p+q)^p = (q-p)^(2p-1) then pq is equals to
Solution
Given the equation:
where and are prime numbers, we want to find the value of .
Step 1: Analyzing the equation
Let's start by analyzing the equation given:
Since and are primes, let's consider different cases to understand how the expression behaves. Notice that for the right-hand side to be defined and meaningful, must be positive (i.e., ).
Step 2: Trying specific small prime values
Let's try the smallest prime number and check the behavior of the equation for different prime .
Case 1:
Substituting into the equation:
Let's find a prime number such that this equation holds.
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Try :
This does not work.
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Try :
This does not work either.
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Try :
This also does not work.
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Try :
This does not work either.
Step 3: Checking if there are other valid cases
None of these small values satisfy the equation. Hence, we try the possibility of checking where the expression is simplified to keep only positive values for both sides. Since for all and chosen above do not hold, we need to analyze for what choice and hold, ensuring both sides of the expression equate.
Step 4: Letting
This assumption helps maintain positivity and equalization on both sides. We test by substituting , keeping and checking with large numbers. If it fails, we conclude that only .
Conclusion
Since only possible value is 2:
Hence, the expression becomes undefined for other primes.
Thus, the correct value of .
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Math Problem Analysis
Mathematical Concepts
Prime numbers
Exponentiation
Equations
Formulas
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Theorems
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Suitable Grade Level
Grades 11-12
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