Question

the sum of a natural number and its reciprocal is 65 by 8 find the numbers

Answer 1

Before I do this, let me tell you to be more precise next time, considering that even fractions can be positive numbers.
However since positive numbers mainly gives me a vibe that you are talking about positive integers, I'll consider that as such.

Let the positive integer be x.
Reciprocal of x= 1/x
By the condition,
x+1/x=65/8
=> (x2​+1)/x=65/8
=>8x2​+8=65x
=>8x2-65x+8=0
=>8x2-64x-x+8=0
=>8x(x-8)-(x-8)=0
=>(x-8)(8x-1)=0
Either x-8=0
=>x=8
Or 8x-1=0
=> x= 1/8

Since x is a positive integer, x=8.

Hope this is right! :)

Answer 2

Let the natural number be “x”

According to question
x + 1/x = 65/8
x^2 + 1 = 65x/8
8x^2 - 65x + 8 = 0
Solve this polynomial equation
We will get x = 8 and x = 1/8
1/8 is not a natural number.
Hence number is 8