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In how many different ways can $$\frac{2}{15}$$ be represented as $$\frac{1}{a} + \frac{1}{b}$$, if $$a$$ and $$b$$ are positive integers with $$a \ge b$$?

I chose to use Simon's Favorite Factoring trick.

I let 2/15 = 1/a + 1/b. Multiplied both sides by 15/2 and ab: ab = 15/2*(a) + 15/2*(b).

Subtracted 15/2*(a) + 15/2*(b) from both sides: ab - 15/2*(a) - 15/2*(b) = 0. Then added 225/4: ab - 15/2*(a) - 15/2(b) + 225/4 = 225/4. I factored the left side: (a - 15/2)*(b - 15/2) = 225/4. And finally, I'm stuck on what to do next. :(

Jun 21, 2018

#1
+101344
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2/15  =

1/8  + 1 / 120

1/ 9  +   1 / 45

1/10  +  1  /30

1/ 12  + 1 / 20

I can generate the first through something called "Egyptian Fractions"....I don't know how the other three are obtained

You can read here : http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html

Jun 21, 2018
#2
+35
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Ummm, ok thanks. Though I think it would've been better to multiply both sides by four. And then, since $$225=3^2\cdot5^2$$, there are $$3\cdot3=9$$ positive integer divisors of $$225$$. Of these, $$\boxed{5}$$ yield ordered pairs of divisors $$(2a-15,2b-15)$$ for which $$a \geq b$$

Toytrain  Jun 21, 2018
#3
+22343
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In how many different ways can
$$\frac{2}{15}$$
be represented as
$$\frac{1}{a} + \frac{1}{b}$$,
if and are positive integers with
$$a \ge b$$ ?

Because $$n = 15$$ is odd:

we calculate $$n^2 = 225$$

And all divisors of $$n^2 = 225$$ are: $$1,3,5,9,15,25,45,75,225$$

$$\text{Let n^2 = p\times q=225  }$$

$$\begin{array}{|r|r|r|r|r|r|r|r|} \hline p & q & s = \frac{p+q}{2} & t = \frac{p-q}{2} & r = \frac{t}{2} & k = \frac{n+\sqrt{n^2+t^2} }{2} & \mathbf{a} = k+r & \mathbf{b} = k-r \\ \hline 225 & 1 & 113 & 112 & 56 & 64 & 120 & 8 \\ 75 & 3 & 39 & 36 & 18 & 27 & 45 & 9 \\ 45 & 5 & 25 & 20 & 10 & 20 & 30 & 10 \\ 25 & 9 & 17 & 8 & 4 & 16 & 20 & 12 \\ 15 & 15 & 15 & 0 & 0 & 15 & 15 & 15 \\ \hline \end{array}$$

$$\begin{array}{|r|r|c|} \hline & \frac{1}{a} + \frac{1}{b} & a \ge b\ \\ \hline 1. & \frac{1}{120} + \frac{1}{8} & \checkmark \\ 2. & \frac{1}{45} + \frac{1}{9} & \checkmark \\ 3. & \frac{1}{30} + \frac{1}{10} & \checkmark \\ 4. & \frac{1}{20} + \frac{1}{12} & \checkmark \\ 5. & \frac{1}{15} + \frac{1}{15} & \checkmark \\ \hline \end{array}$$

Jun 21, 2018