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.

Let unit’s digit = y

and the ten’s digit = x

So, the original number = 10x + y

The sum of the number = 10x + y

The sum of the digit = x + y

According to the question,

⇒ 10x + y = 6(x + y)

⇒ 10x + y = 6x + 6y

⇒ 10x + y – 6x – 6y

⇒ 4x – 5y = 0 …(i)

The reverse of the number = x + 10y

and 10x + y – 9 = x + 10y

⇒ 10x + y – 9 = x + 10y

⇒ 10x – x + y – 10y = 9

⇒ 9x – 9y = 9

⇒ x – y = 1

⇒ x = y + 1 …(ii)

On substituting the value of x = y + 1 in Eq. (i), we get

4x – 5y = 0

⇒ 4(y + 1) – 5y = 0

⇒ 4y + 4 – 5y = 0

⇒ 4 – y = 0

⇒ y = 4

On substituting the value of y = 4 in Eq. (ii), we get

x = y + 1

⇒ x = 4 + 1

⇒ x = 5

So, the Original number = 10x + y

= 10×5 + 4

= 50 + 4

= 54

Hence, the two digit number is 54.