The sum of a natural number and its reciprocal is 65/8. Find the number.

Let the required natural numbers x

According to given condition,

Using the splitting middle term – the middle term of the general equation is divided in two such values that:

Product = a.c

For the given equation a = 8 b = – 65 c = 8

= 8.8 = 64

And either of their sum or difference = b

= – 65

Thus the two terms are – 64 and – 1

Difference = – 64 – 1 = – 65

Product = – 64. – 1 = 64

8x (x – 8) – 1(x – 8) = 0

(x – 8) (8x – 1) = 0

(x – 8) = 0 or (8x – 1) = 0

x = 8 or x = 1/8

x = 8 (x is a natural number)

Hence the required number is 8.

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