Determine all possible positive integers m such that:
√(4m - 7)
...........
√(m + 1)
is a rational number.
Provide adequate reasoning for your answer.
The square roots are a help here for once.
Given that sqrt(4m-7)/sqrt(m+1) is rational for some positive integer m, choose positive integers a,b such that:
a sqrt(4m-7) = b sqrt(m+1)
But since a and b are both integers, sqrt(4m-7) and sqrt(m+1) must both also be integers, such that:
(4m-7)=x^2, (m+1)=y^2, equivalently
(4m-7)=x^2, 4(m+1) = (2y)^2
Then
(2y)^2-x^2=11, equivalently
(2y-x)(2y+x)=11
11 is prime, so 2y-x=1, giving y=3
4(m+1) = (2*3)^2, so m=8, the sole solution.
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Posted by broll
on 2023-02-22 22:41:08 |
A different method
