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

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

Let us consider, ones digit of a two digit number = x and

Tens digit = y

The number is x + 10y

By reversing the digits,

One’s digit = y and ten’s digit = x

The number is y + 10x

As per the statement,

(x + 10y)/(x+y) = 6

x + 10y = 6(x + y)

5x = 4y

or x = 4/5 y …………(1)

And,

x + 10y – 9 = y + 10x

x + 10y – y – 10x = 9

-9x + 9y = 9

x – y = -1

using equation (1)

4/5 y – y = -1

y = 5

From (1): x = 4/5(5) = 4

Answer:

The number is: x + 10y = 4 + 10(5) = 54