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Olympiad workbook
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Olympiad workbook
Subject
IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

If x and y are positive integers such that (x-4) (x-10)=2^y , find...

If xx and yy are positive integers such that (x−4)(x−10)=2y(x-4) (x-10)=2^y, find the maximum possible value of x+yx+y.

Correct answer

16

Step-by-step solution

 Let y=a+b(x−4)(x−10)=2a⋅2b∴x=2a+4 or 2b+10 (vice-versa). ∴x=2a+4=2b+10.\begin{aligned} & \text { Let } y=a+b \\ & (x-4)(x-10)=2^a \cdot 2^b \\ & \therefore \quad x=2^a+4 \text { or } 2^b+10 \text { (vice-versa). } \\ & \therefore \quad x=2^a+4=2^b+10 . \end{aligned} ⇒2a−2b=6.⇒a=3 and b=1.∴y=4 and x=12⇒x+y=16.\begin{aligned} & \Rightarrow 2^a-2^b=6 . \\ & \Rightarrow a=3 \text { and } b=1 . \quad \therefore \quad y=4 \text { and } x=12 \\ & \Rightarrow x+y=16 .\end{aligned}
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