A problem from the Egyptian Olympiad

Number Theory Level 4

How many positive integers ordered pairs ( x , y ) (x,y) satisfy the following relation:

1 x + 1 y = 1 4 \frac 1x + \frac 1y = \frac 14

1 3 4 5

Abdulrahman Saber Abdel-azeem by Abdulrahman Saber Abdel-… , 19

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1 solution

Department 8
Nov 22, 2018

1 x + 1 y = 1 4 4 ( x + y ) = x y ⇒ ( x − 4 ) ( y − 4 ) = 16 \frac{1}{x}+\frac{1}{y}=\frac{1}{4} \\ 4(x+y)=xy\Rightarrow (x-4)(y-4)=16

Now since we are looking for positive integers, express 16 16 as multiplication of two numbers and equal them x − 4 x-4 and y − 4 y-4 individually. BTW the through the questions statements the answer is: 9 9 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

(8,8) , (5,20) , (20,5) , (12,6) , (6,12) is there any other ordered pairs ?

Abdulrahman Saber Abdel-azeem - 2 years, 6 months ago

I've just noticed that! thank you, I meant "ordered pairs" , it was my mistake.

Abdulrahman Saber Abdel-azeem - 2 years, 6 months ago

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