Question

A two digit number is such that the product of the digit is 12. When 36 is added to the number, the digits interchange their places. Find the numbers.

Answer 1

Let the two digit number be 10x + y, where:

x = tens digit

y = units digit

The product of digits = 12

xy = 12,    ...(i) 

When 36 is added, digits interchange places.

10x + y + 36 = 10y + x

10x − x + y − 10y = −36

⇒ 9x − 9y + 36 = 0   ...(Both side divided by 9)

⇒ x − y = −4

⇒ y − x = +4    ...(ii)

Putting xy = 12 from (i) in (ii), we get

xy = 12,    

y − x = 4   ...(iii)

Substitute y = x + 4 into xy = 12

x(x + 4) = 12

x² + 4x − 12 = 0

⇒ x² + 6x − 2x − 12 = 0 

x(x + 6) −2(x + 6) = 0

(x − 2)(x + 6) = 0

x − 2 = 0 or x + 6 = 0

x = 2 or x = −6

∴ x = −6 (not allowed; digit can't be negative)

So x = 2

Putting x = 2 in eq. no. (iii)

y = x + 4 

= 2 + 4 

= 6

Putting x and y value in the given equation

= 10x + y

= 10 × 2 + 6

= 26

Hence the number is 26.