A two-digit number is such that the product of its digits is 35. If 18

1.	A two-digit number is such that the product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the number.
Answer» Let the tens and the units digits of the required number be x and y, respectively. Then, we have: xy = 35 …….(i) Required number = (10x + y) Number obtained on reversing its digits = (10y + x) ∴(10x + y) + 18 = 10y + x ⇒9x – 9y = -18 ⇒ 9(y – x) = 18 ⇒ y – x = 2 ……..(ii) We know: (y + x)²– (y – x)² = 4xy ⇒ (y + x) = ± √( x− y) ² + 4 ⇒ (y + x) = ± √4 + 4 ×35 = ± √144 = ±12 ⇒ y + x = 12 ……..(iii) (∵ x and y cannot be negative) On adding (ii) and (iii), we get: 2y = 2 +12 = 14 ⇒y = 7 On substituting y = 7 in (ii) we get 7 – x = 2 ⇒ x = (7 – 2) = 5 ∴ The number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57 Hence, the required number is 57.

A two-digit number is such that the product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the number.

Answer»

Let the tens and the units digits of the required number be x and y, respectively.

Then, we have:

xy = 35 …….(i)

Required number = (10x + y)

Number obtained on reversing its digits = (10y + x)

∴(10x + y) + 18 = 10y + x

⇒9x – 9y = -18

⇒ 9(y – x) = 18

⇒ y – x = 2 ……..(ii)

We know:

(y + x)²– (y – x)² = 4xy

⇒ (y + x) = ± √( x− y) ² + 4

⇒ (y + x) = ± √4 + 4 ×35 = ± √144 = ±12

⇒ y + x = 12 ……..(iii) (∵ x and y cannot be negative)

On adding (ii) and (iii), we get:

2y = 2 +12 = 14

⇒y = 7

On substituting y = 7 in (ii) we get

7 – x = 2

⇒ x = (7 – 2) = 5

∴ The number = (10x + y) = 10 × 5 + 7 = 50 + 7 = 57

Hence, the required number is 57.