A two - digit num

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 = 18

⇒ x = 18/y.....(1)

and,

(10x + y) - 63 = (10y + x)

⇒ 9x - 9y = 63

⇒ x - y = 7.....(2)

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

⇒ 18 - y² = 7y

⇒ y² + 7y - 18 = 0

⇒ y² + 9y - 2y - 18 = 0

⇒ y(y + 9) - 2(y + 9) = 0

⇒ (y + 9)(y - 2) = 0

∴ y = 2

[y = - 9 is invalid because digits of a number cannot be negative.]

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

x = 9

Thus, the required number is 92.