A number consists of two digits. When it is divided by the sum of its

1.	A number consists of two digits. When it is divided by the sum of its digits, the quotient is 6 with no remainder. When the number is diminished by 9, the digits are reversed. Find the number.
Answer» We know: Dividend = Divisor × Quotient + Remainder Let the tens and the units digits of the required number be x and y, respectively. Required number = (10x + y) 10x + y = (x + y) × 6 + 0 ⇒10x – 6x + y – 6y = 0 ⇒ 4x – 5y = 0 …….(i) Number obtained on reversing its digits = (10y + x) ∴ 10x + y - 9 = 10y + x ⇒9x – 9y = 9 ⇒x – y = 1 ……..(ii) On multiplying (ii) by 5, we get: 5x – 5y = 5 ……..(iii) On subtracting (i) from (iii), we get: x = 5 On substituting x = 5 in (i) we get 4 × 5 – 5y = 0 ⇒ 20 - 5y = 0 ⇒ y = 4 ∴ The number = (10x + y) = 10 × 5 + 4 = 50 + 4 = 54 Hence, the required number is 54.

A number consists of two digits. When it is divided by the sum of its digits, the quotient is 6 with no remainder. When the number is diminished by 9, the digits are reversed. Find the number.

Answer»

We know:

Dividend = Divisor × Quotient + Remainder

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

Required number = (10x + y)

10x + y = (x + y) × 6 + 0

⇒10x – 6x + y – 6y = 0

⇒ 4x – 5y = 0 …….(i)

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

∴ 10x + y - 9 = 10y + x

⇒9x – 9y = 9

⇒x – y = 1 ……..(ii)

On multiplying (ii) by 5, we get:

5x – 5y = 5 ……..(iii)

On subtracting (i) from (iii), we get:

x = 5

On substituting x = 5 in (i) we get

4 × 5 – 5y = 0

⇒ 20 - 5y = 0

⇒ y = 4

∴ The number = (10x + y) = 10 × 5 + 4 = 50 + 4 = 54

Hence, the required number is 54.