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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 decreased by 9, the digits are reversed. What is the number?

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answer is B.

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Detailed Solution

Given that, when a number is divided by the sum of its digits, the quotient is 6 with no remainder.
When the number is decreased by 9, the digits are reversed.
Let the unit’s digit be y and the ten’s digit be x.
Then, the original number be

10 x + y .

After reversing the digits, the new number =

x + 10 y

According to the given information, when a number is divided by the sum of its digits, the quotient is 6 with no remainder.

10 x + y x + y = 6 ⇒ 10 x + y = 6 x + y ⇒ 10 x + y = 6 x + 6 y ⇒ 10 x − 6 x + − 6 y + y = 0 ⇒ 4 x − 5 y = 0 ... … 1

When the number is diminished by 9, the digits are reversed.

10 x + y − 9 = x + 10 y ⇒ 9 x − 9 y − 9 = 0 ⇒ x − y − 1 = 0 ⇒ x − y = 1.

⇒ x = y + 1 ...... (2)

Substituting the value of

x = y + 1

in the first equation and solving it, we get,

4 x − 5 y = 0 ⇒ 4 y + 1 − 5 y = 0 ⇒ 4 y + 4 − 5 y = 0 ⇒ − y = 0 − 4 ⇒ y = 4

Putting the value of y = 4 in equation 3 to find the value of x, we get,

x = y + 1 ⇒ x = 4 + 1 ⇒ x = 5.

Original number =

10 x + y

⇒ 10 x + y = 10 (5) + 4 ⇒ 10 x + y = 50 + 4 ⇒ 10 x + y = 54

Hence, the number is 54.
Thus, option 2 is correct.

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