Question

A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.

Answer 1

Let 2-digit number = xy = 10x + y
Reversed digits = yx = 10y + x
According to question,
xy = 6

y = `(6)/x`       ...(i)
and
10x + y + 9 = 10y + x

⇒ `10x + (6)/x + 9 = 10 xx (6)/x + x    ...("From" (i) y = (6)/x)`

⇒ 10x² + 6 + 9x = 60 + x²
⇒ 10x² - x² + 9x + 6 - 60 = 0
⇒  9x² + 9x - 54 = 0
⇒ x² + x - 6 = 0
⇒ x² + 3x - 2x - 6 = 0
⇒ x(x + 3) -2(x + 3) = 0
⇒ x = 2 or -3  ...(rejacting -3)
putting the value of x in (i)

y = `(6)/(2)` = 3
∴ 2-digit
= 10x + y
= 10 x 2 + 3
= 23.