Question

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 1

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.