Question

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

Answer 1

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 ri8^_^

Answer 2

• Let the natural number be M.

And

• Reciprocal of the natural number = $\dfrac{1}{M}$

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

A.T.Q.

=> M + $\dfrac{1}{M}$ = $\dfrac{65}{8}$

=> $\dfrac{{M}^{2}\:+\:1}{M}$ = $\dfrac{65}{8}$

Cross-multiply them..

=> 8(M² + 1) = 65M

=> 8M² + 8 = 65M

=> 8M² - 65M + 8 = 0

=> 8M² - 64M - M + 8 = 0

=> 8M(M - 8) -1(M - 8) = 0

=> (M - 8) (8M - 1) = 0

• M - 8 = 0

=> M = 8

• 8M - 1 = 0

=> 8M = 1

=> M = 1/8

_______________________________

Natural number is 8.

____________ [ANSWER ]

_______________________________