Question

Show that there are no natural numbers m and n such that 7/17 = (1/m) + (1/n)

Answer 1

Taking RHS,

(1/m)+(1/n)

=>(m + n)/(m x n)

Now taking the LHS,

7/17

we know 17 is a prime and only possible multiples of 17 are 1 and 17.

So, from LHS and RHS comparing the denominator we can deduce:

m x n = 1 x 17

or m x n = 17 x 1

Now according to our observation

m + n = 17 + 1

But from comparing numerators of given equation

m + n should give 7

which is not possible for any pair of natural numbers.

Hence, there are no natural numbers m and n such that 7/17 = (1/m) + (1/n)