Prove by contradiction that there are no positive integers p and q such that
$$4p^2 - q^2 = 25$$ - Edexcel - A-Level Maths Pure - Question 1 - 2020 - Paper 1
Question 1
Prove by contradiction that there are no positive integers p and q such that
$$4p^2 - q^2 = 25$$
Worked Solution & Example Answer:Prove by contradiction that there are no positive integers p and q such that
$$4p^2 - q^2 = 25$$ - Edexcel - A-Level Maths Pure - Question 1 - 2020 - Paper 1
Step 1
Set up the contradiction and factorise
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Answer
Assume that there exist positive integers p and q such that:
(2p+q)(2p−q)=25
Given that 25 is an odd number, both factors must be odd. Therefore, we can express the factors as:
Let 2p+q=25 and 2p−q=1.
Now, we have two equations to work with.
Step 2
Solve for p and q
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Answer
From the equation 2p+q=25, we can rearrange it to find q:
q=25−2p
Substituting this expression for q into the second equation:
2p−(25−2p)=1
Simplifying gives:
2p−25+2p=1
4p−25=1
Thus, we find:
p = 6.5$$
Since p is not a positive integer, this case provides a contradiction.
Step 3
Consider the alternative factorization
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Answer
Now, consider the second case:
Let 2p+q=5 and 2p−q=5.
From 2p+q=5, we can isolate q:
q=5−2p
Substituting q into the second equation:
2p−(5−2p)=5
This simplifies to:
2p−5+2p=5
4p = 10 \
p = 2.5$$
Again, since p is not a positive integer, we reach another contradiction.
Step 4
Conclusion
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Answer
Since both cases lead us to a contradiction where p is not a positive integer, we conclude that:
