Positive integers are cool

Number Theory Level 3

Let $x$ and $y$ be positive integers, with $x<y$ , such that the following equation holds:

$\large 2xy = (x+4)(y+4)$

Find the sum of all possible values of $x$ .

The answer is 19.

2 solutions

David Vreken
Aug 4, 2019

The equation $2xy = (x + 4)(y + 4)$ rearranges to $(x - 4)(y - 4) = 32$ , and the factors of $32$ so that $x - 4 < y - 4$ are $x - 4 = 1$ , $x - 4 = 2$ , and $x - 4 = 4$ , which means $x = 5$ , $x = 6$ , and $x = 8$ , and $5 + 6 + 8 = \boxed{19}$ .

Denton Young
Aug 2, 2019

$2xy = (x+4)(y+4)$

$xy = 4x + 4y + 16$

$x(y-4) =4(y+4)$

$x = 4(1 + (8/(y-4)))$

y = 2, 3, 5, 6, 8, 12, 20 or 36.

The only possibilities that have x < y are (x = 5, y = 36), (x = 6, y= 20) and (x = 8, y = 12.)

No idea how you made the leap from step3 to step4

Malcolm Rich - 1 year, 10 months ago

Divided both sides by (y-4) and simplified.

Denton Young - 1 year, 10 months ago

In line four, the denominator should be $y - 4$ , rather than $y + 4$ .

$x = 4 \Big( 1 + \frac {8}{y - 4} \Big)$

Matthew Feig - 1 year, 10 months ago

My mistake. Fixed, thanks.

Denton Young - 1 year, 10 months ago

You write (y+4) as (y-4+8). :)

Donát Herczeg - 1 year, 7 months ago

Positive integers are cool

The answer is 19.

2 solutions

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