Let the two - digit number be xy (i.e. 10x + y).

After reversing the digits of the number xy, the new number becomes yx (i.e. 10y + x).

According to question -

xy = 35

⇒ x = 35/y.....(1)

and,

(10x + y) + 18 = (10y + x)

⇒ 9x - 9y = - 18

⇒ x - y = - 2.....(2)

Substituting the value of x in equation (2), we get -

⇒ 35 - y² = - 2y

⇒ y² - 2y - 35 = 0

⇒ y² - 7y + 5y - 35 = 0

⇒ y(y - 7) + 5(y - 7) = 0

⇒ (y + 5)(y - 7) = 0

∴ y = 7

[y = - 5 is invalid because digits of a number cannot be negative.]

Substituting the value of y in equation (1), we get -

x = 5

Thus, the required number is 57.