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# help plz

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need help with fractions plz

Given positive integers $x$ and $y$ such that $x\neq y$ and $\frac{1}{x} + \frac{1}{y} = \frac{1}{6}$, what is the smallest possible value for $x + y$?

Aug 15, 2023

#1
+121
+1

Let's work with the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{6}$$.

First, we can clear the fractions by multiplying both sides of the equation by $$6xy$$:

$6y + 6x = xy.$

Rearrange the terms:

$xy - 6x - 6y = 0.$

Now, we can use Simon's Favorite Factoring Trick to factor this equation:

$xy - 6x - 6y + 36 = 36.$
$(x - 6)(y - 6) = 36.$

We are looking for two distinct positive integer solutions $$x$$ and $$y$$ such that $$(x - 6)(y - 6) = 36$$.

The pairs of factors of 36 are: $$(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)$$.

If we let $$x - 6 = 1$$ and $$y - 6 = 36$$, we get $$x = 7$$ and $$y = 42$$, which doesn't satisfy $$x \neq y$$.

If we let $$x - 6 = 2$$ and $$y - 6 = 18$$, we get $$x = 8$$ and $$y = 24$$.

So, the solution with the smallest possible $$x + y$$ is when $$x = 8$$ and $$y = 24$$, which gives $$x + y = 8 + 24 = 32$$.

Therefore, the smallest possible value for $$x + y$$ is $$32$$.

Aug 15, 2023

#1
+121
+1

Let's work with the equation $$\frac{1}{x} + \frac{1}{y} = \frac{1}{6}$$.

First, we can clear the fractions by multiplying both sides of the equation by $$6xy$$:

$6y + 6x = xy.$

Rearrange the terms:

$xy - 6x - 6y = 0.$

Now, we can use Simon's Favorite Factoring Trick to factor this equation:

$xy - 6x - 6y + 36 = 36.$
$(x - 6)(y - 6) = 36.$

We are looking for two distinct positive integer solutions $$x$$ and $$y$$ such that $$(x - 6)(y - 6) = 36$$.

The pairs of factors of 36 are: $$(1, 36), (2, 18), (3, 12), (4, 9), (6, 6)$$.

If we let $$x - 6 = 1$$ and $$y - 6 = 36$$, we get $$x = 7$$ and $$y = 42$$, which doesn't satisfy $$x \neq y$$.

If we let $$x - 6 = 2$$ and $$y - 6 = 18$$, we get $$x = 8$$ and $$y = 24$$.

So, the solution with the smallest possible $$x + y$$ is when $$x = 8$$ and $$y = 24$$, which gives $$x + y = 8 + 24 = 32$$.

Therefore, the smallest possible value for $$x + y$$ is $$32$$.

SpectraSynth Aug 15, 2023