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A two-digit number is such that the product of its digits is 12. When 9 is added to the number, the digits are interchanged. Find the number.

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-43

both (1) and (2)

answer is C.

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Detailed Solution

It is given that a product of two-digit number is 12 and when 9 is added to the number, the digits are interchanged.
Let us consider the digits of two-digit number are x and y, then number is (10x+y).
Given product of digits is 12. So,

x × y = 12

........... (1)
Also given, when 9 is added to the number, the digits are interchanged.

10 x + y + 9 = 10 y + x

⇒ 9 x − 9 y = − 9

⇒ x − y = − 1

............ (2)
Substitute the value of y in equation (2) from equation (1),

x − 12 x = − 1 ⇒ x 2 − 12 = − x ⇒ x 2 + x − 12 = 0 ⇒ x 2 + 4 x − 3 x − 12 = 0

On further simplifying,

⇒ x x + 4 − 3 x + 4 = 0 ⇒ x + 4 x − 3 = 0 ⇒ x = 3, − 4

From equation (1),

y = 12 x ⇒ y = 123, 12 − 4 ⇒ y = 4, − 3

The set of values for (x, y) are

3, 4, − 4, − 3

.
Consider the number is

10 x + y

.
Consider the first set of value for digits

3, 4

= 10 x + y = 10 × 3 + 4 = 34

Consider the set of value for digits

− 4, − 3

= 10 x + y = 10 × − 4 + − 3 = − 43

The numbers are

34, − 43

.
The correct option is 3).

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