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.