Find all pairs of positive integers x and y such that 2x+1 is divisible by y and at the same time 2y+1 is divisible by x.Other solutions and proofs that I have seen did not actually give all the...

Find all pairs of positive integers x and y such that 2x+1 is divisible by y and at the same time 2y+1 is divisible by x.

Other solutions and proofs that I have seen did not actually give all the possible answers or explain that the solutions they have given are the ONLY possible answers. 

Assistance would be greatly appreciated, as would a comprehensive and detailed proof outlining everything clearly. 

Thanks!

Asked on by lennywolff

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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Since `2x + 1`  is divisible by y, hence, dividing `2x+1`  by y yields a reminder equal to 0 such that:

`(2x+1)/y = k => ky = 2x + 1 => 2x = ky - 1 => x = (ky-1)/2`

k is integer

Since x is an integer, hence `(ky-1)`  is a multiple of 2.

Since `2y + 1`  is divisible by x, hence, dividing `2y+1`  by x yields a reminder equal to 0 such that:

`(2y+1)/x = p => px = 2y + 1 => 2y = px - 1 => y = (px - 1)/2`

p is integer

`y = (p(ky-1)/2 - 1)/2 => y = (pky - p - 2)/4 => 4y - pky = -p - 2`

Factoring out y yields:

`y(pk - 4) = p + 2 => y = (p + 2)/(pk - 4)`

Substituting `(p + 2)/(pk - 4)`  for y in `x = (ky-1)/2`  yields:

`x = (k(p + 2)/(pk - 4) - 1)/2 => x = (pk + 2k - pk + 4)/(2(pk-4))`

`x = 2(k+2)/(2(pk-4)) => x = (k+2)/(pk-4)`

Hence, evaluating the integers x and y, under the given conditions, yields `x = (k+2)/(pk-4)`  and `y = (p + 2)/(pk - 4), ` where `(p + 2)`  is a divisor of `k+2`  and `p+2.`

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