Question

Given positive integers x and y such that x not = y and 1/x + 1/y = 1/12, what is the smallest possible positive value for x + y?

Answer 1

Answer:

x = 21 or 28

y = 28 or 21

Step-by-step explanation:

I cannot tell you why this is true. It is certainly beyond intuition. The answer is 49 is the sum. The two numbers making the sum are 21 and 28.

If you take out the restriction of x<>y then the minimum sum is both x and y = 24. You might be able to get that using a quadratic formula.

I think you can 12*2x = x^2

x^2 = 24x Divide by x.

x = 24

The sum of xy= 48.

The formula for this is

1/x + 1/y    = 1/12     Put the left over a common denominator

(x + y)/xy = 1/12       Multiply both sides by 12 xy

12*(x + y) =  xy       This is called a Diophantine equation where it is not reducible any further, but there are integer solutions.

The largest pair I found letting x and y both check to 10000 was

x = 13 y = 156  and

x = 156 y = 13