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#### eddybob123

##### Active member

- Aug 18, 2013

- 76

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{8}\).

2) Generalize to

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{n}\).

If no one gets the answer in long time I'll post hints.

- Thread starter eddybob123
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- Thread starter
- #1

- Aug 18, 2013

- 76

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{8}\).

2) Generalize to

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{n}\).

If no one gets the answer in long time I'll post hints.

- Feb 9, 2012

- 33

Hi there eddybob123 , welcome to MHB !

-agentredlum

-agentredlum

- Mar 10, 2012

- 834

For the generalized part put $a=n-x$ and $b=n-y$. One gets $n^2=xy$.

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{8}\).

2) Generalize to

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{n}\).

If no one gets the answer in long time I'll post hints.

- Jan 25, 2013

- 1,225

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{n}\).

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{8}\).

2) Generalize to

\(\displaystyle \frac{1}{a}+\frac{1}{b}=\frac{1}{n}\).

If no one gets the answer in long time I'll post hints.

$\dfrac{1}{n}=\dfrac {k}{nk}=\dfrac {1+(k-1)}{nk}=\dfrac {1}{nk}+\dfrac {k-1}{nk}$ (k=2,3,----)

here nk must be a multiple of (k-1)

take n=8

$\dfrac {1}{8}=\dfrac {1}{16}+\dfrac{1}{16}=\dfrac {1}{24}+\dfrac{1}{12}=

\dfrac {1}{40}+\dfrac{1}{10}=\dfrac {1}{72}+\dfrac{1}{9}$

I found four pairs of (a,b)

Last edited:

- Jan 25, 2013

- 1,225

$n=xy ,\,\, x\geq 1,\,\, y\geq 1$

$\dfrac {1}{n}=\dfrac {k}{nk}=\dfrac {1+(k-1)}{nk}=\dfrac{1}{nk}+\dfrac{k-1}{nk}$

here nk mod (k-1)=0-----(1)

$k=x,y,(x+y),---$

$if \,\, n=8=1\times 8=2\times 4$

$\therefore k=2,4,(2+4),(1+8),(2+4+1+8)----(2)$

compare (1) and (2) and choose suitable k,and get corresponding a and b

$\dfrac {1}{n}=\dfrac {k}{nk}=\dfrac {1+(k-1)}{nk}=\dfrac{1}{nk}+\dfrac{k-1}{nk}$

here nk mod (k-1)=0-----(1)

$k=x,y,(x+y),---$

$if \,\, n=8=1\times 8=2\times 4$

$\therefore k=2,4,(2+4),(1+8),(2+4+1+8)----(2)$

compare (1) and (2) and choose suitable k,and get corresponding a and b

Last edited:

- Thread starter
- #6

- Aug 18, 2013

- 76

Hi there eddybob123 , welcome to MHB !

-agentredlum