A two-digit number is such that the product of the digits is 16. When

1.	A two-digit number is such that the product of the digits is 16. When 54 is subtracted from the number, the digits are interchanged. Find the number.
Answer» Let the ones digit be ‘a’ and tens digit be ‘b’. Given, two-digit number is such that the product of its digits is 16. ⇒ ab = 16 --- (1) Also, when 54 is subtracted from the number, the digits interchange their places ⇒ 10b + a – 54 = 10a + b ⇒ 9b – 9a = 54 ⇒ b – a = 6 ⇒ b = 6 + a Substituting in 1 ⇒ a × (6 + a) = 16 ⇒ a²+ 6a – 16 = 0 ⇒ a² + 8a – 2a – 16 = 0 ⇒ a(a + 8) – 2(a + 8) = 0 ⇒ (a – 2)(a + 8) = 0 ⇒ a = 2 Thus, b = 8 Number is 82

A two-digit number is such that the product of the digits is 16. When 54 is subtracted from the number, the digits are interchanged. Find the number.

Answer»

Let the ones digit be ‘a’ and tens digit be ‘b’.

Given, two-digit number is such that the product of its digits is 16.

⇒ ab = 16 --- (1)

Also, when 54 is subtracted from the number, the digits interchange their places

⇒ 10b + a – 54 = 10a + b

⇒ 9b – 9a = 54

⇒ b – a = 6

⇒ b = 6 + a

Substituting in 1

⇒ a × (6 + a) = 16

⇒ a²+ 6a – 16 = 0

⇒ a² + 8a – 2a – 16 = 0

⇒ a(a + 8) – 2(a + 8) = 0

⇒ (a – 2)(a + 8) = 0

⇒ a = 2

Thus, b = 8

Number is 82