The number of solutions of 1x+1y=16 where x,y∈N

1x+1y=16⇒6(x+y)=xy⇒(x−6)(y−6)=36⇒(x−6)∣36.⇒x−6=1,2,3,4,6,9,12,18,36⇒(x,y)=(7,42),(8,24),(9,18),(10,15),(12,12), (15,10),(18,9),(24,8),(42,7)

The number of solutions of 1x+1y=16 where x,y∈N

1x+1y=16⇒6(x+y)=xy⇒(x−6)(y−6)=36⇒(x−6)∣36.⇒x−6=1,2,3,4,6,9,12,18,36⇒(x,y)=(7,42),(8,24),(9,18),(10,15),(12,12), (15,10),(18,9),(24,8),(42,7)

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The number of solutions of $\frac{1}{x} + \frac{1}{y} = \frac{1}{6}$ where $x, y \in N$

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Detailed Solution

$\begin{array}{l} \frac{1}{x} + \frac{1}{y} = \frac{1}{6} \Rightarrow 6 (x + y) = x y \\ \Rightarrow (x - 6) (y - 6) = 36 \\ \Rightarrow (x - 6) ∣ 36 . \\ \Rightarrow x - 6 = 1, 2, 3, 4, 6, 9, 12, 18, 36 \\ \Rightarrow (x, y) = (7, 42), (8, 24), (9, 18), (10, 15), (12, 12), \\ (15, 10), (18, 9), (24, 8), (42, 7) \end{array}$