Let the tens and the units digits of the required number be x and y, respectively.
Then, we have:
xy = 18 …….(i)
Required number = (10x + y)
Number obtained on reversing its digits = (10y + x)
∴(10x + y) - 63 = 10y + x
⇒9x – 9y = 63
⇒ 9(x – y) = 63
⇒ x – y = 7 ……..(ii)
We know:
(x + y)2 – (x – y)2 = 4xy
⇒ x + y = 11 ……..(iii) (∵ x and y cannot be negative)
On adding (ii) and (iii), we get:
2x = 7 +11 = 18
⇒x = 9
On substituting x = 9in (ii) we get
9 – y = 7
⇒ y = (9 – 7) = 2
∴ Number = (10x + y) = 10 × 9 + 2 = 90 + 2 = 92
Hence, the required number is 92.