A two digit number is such that the product of its digits, is 8. When

1.	A two digit number is such that the product of its digits, is 8. When 18 is added to the number, they interchange their places. Determine the number.
Answer» Let the digit in the units place = x Let the digit in the tens place = y ∴ The number = 10y + x By interchanging the digits the number becomes 10x + y By problem (10x + y) – (10y + x) = 18 ⇒ 9x – 9y = 18 ⇒ 9(x – y) =18 ⇒ x – y = 18/9 = 2 ⇒ y = x – 2 (i.e.) digit in the tens place = x – 2 digit in the units place = x Product of the digits = (x – 2) x By problem x² – 2x = 8 x² – 2x – 8 = 0 ⇒ x² – 4x + 2x – 8 = 0 ⇒ x(x – 4) + 2(x – 4) = 0 ⇒ (x – 4) (x + 2) = 0 ⇒ x – 4 = 0 (or) x + 2 = 0 ⇒ x = 4 (or) x = -2 ∴ x = 4 [∵ x can’t be negative] ∴ The number is 24.

A two digit number is such that the product of its digits, is 8. When 18 is added to the number, they interchange their places. Determine the number.

Answer»

Let the digit in the units place = x

Let the digit in the tens place = y

∴ The number = 10y + x

By interchanging the digits the number becomes 10x + y

By problem (10x + y) – (10y + x) = 18

⇒ 9x – 9y = 18

⇒ 9(x – y) =18

⇒ x – y = 18/9 = 2

⇒ y = x – 2

(i.e.) digit in the tens place = x – 2

digit in the units place = x

Product of the digits = (x – 2) x

By problem x² – 2x = 8

x² – 2x – 8 = 0

⇒ x² – 4x + 2x – 8 = 0

⇒ x(x – 4) + 2(x – 4) = 0

⇒ (x – 4) (x + 2) = 0

⇒ x – 4 = 0 (or) x + 2 = 0

⇒ x = 4 (or) x = -2

∴ x = 4 [∵ x can’t be negative]

∴ The number is 24.