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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.

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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.

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