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A two-digit number is such that the product of its digits is 15. If 18 is added to the number, the digits interchange their places. Find the number.

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answer is C.

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Detailed Solution

It is given that a two-digit number is such that the product of its digits is 15.
It is also given that if 18 is added to the number, the digits interchange their places.
Let the digits of the number at tens place

x

and at unit place be

y

.
Therefore, the given number is

10 x + y

.
As the product of the digits of the number is 15,

⇒ x y = 15 ⇒ y = 15 x ... (1)

If 18 is Added to the number. The digits interchange their places.

⇒ 10 x + y + 18) = 10 y + x ⇒ 9 x − 9 y = − 18 ⇒ x − y = − 2... (2)

Substituting (1) in equation (2) to determine the value of

x

⇒ x − y = − 2 ⇒ x − 15 x = − 2 ⇒ x 2 + 2 x − 15 = 0 ⇒ x 2 + 5 x − 3 x − 15 = 0 ⇒ x (x + 5) − 3 (x + 5) = 0 ⇒ (x − 3) (x + 5) = 0

Solving these factors by equating both the terms to 0,

⇒ x − 3 = 0 ⇒ x = 3

And,

⇒ x + 5 = 0 ⇒ x = − 5

Therefore,

x = 3, − 5

.
Since the digits are not negative,

x = 3

.
Substitution 3 for

x

in equation (1) to determine the value of

y

⇒ y = 15 x ⇒ y = 153 ⇒ y = 5

Then the number is,

= 10 x + y = 10 (3) + 5 = 35

The number is 35.
Hence, the correct option is 3.

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