Question

A two-digit number is such that the product of its digits is 14 .if 45is added to the number the digits interchange their places.find the number​

Answer 1

Answer:

the answer is 27.

Step-by-step explanation:

let the two no. are X and y.

then x × y = 14 equation 1...

y = 14/x

now according to the questions

( 10 X + y) + 45 = ( 10 y + x )

( 10 X + 14/x ) + 45 = ( 10 × 14/x + x)

( 10x² + 14 + 45x ) = ( 140 + x² )

9x² + 45x -126=0

x= 2, -7

we only take positive value

so x= 2

y = 14/2

= 7

so the no. are 27

and 27 + 45 = 72.

Answer 2

Answer:

27

or

-72

Step-by-step explanation:

Let the digits of the two digit number be x and y .

So the number = 10x + y

According to Question ,

The product of the digits is 14

Therefore ,

$x \times y = 14 \\ \\ \implies \: x = \frac{14}{y} - - - (1)$

Again , it is given that if 45 is added to the number , the digits interchange there position .

Thus

$10x + y + 45 = 10y + x \\ \\ \implies10x - x + y - 10y + 45 = 0 \\ \\ \implies9x - 9y + 45 = 0 \\ \\ \implies9(x - y + 5) = 0 \\ \\ \implies \: x - y + 5 = 0 - - - (2)$

$using \: x = \frac{14}{y} \: in \: (2) \: we \: have \\ \\ \implies \: \frac{14}{y} - y + 5 = 0 \\ \\ \implies \frac{14 - {y}^{2} + 5y }{y} = 0 \\ \\ \implies14 - {y}^{2} + 5y = 0 \\ \\ \implies {y}^{2} - 5y - 14 = 0 \\ \\ \implies {y}^{2} + 2y - 7y - 14 = 0 \\ \\ \implies \: y(y + 2) - 7(y + 2) = 0 \\ \\ \implies \: (y + 2)(y - 7) = 0$

Therefore , the value of y will be :

$y + 2 = 0 \\ \implies \: y = - 2 \\ \\ and \\ y - 7 = 0 \\ \implies \: y = 7$

Now using the value of y in (1) we have :

$x = \frac{14}{ - 2} \\ \implies \: x = - 7 \\ \\ and \: \\ x = \frac{14}{7} \\ \implies \: x = 2$

When x = 2 and y = 7

The number is

10×2 + 7

= 20 + 7

= 27

Again , when x = -7 and y = -2

The number is

10×(-7) -2

=-70-2

= -72

Verification :

The product of digits

★ First case :

7×2

= 14

★ second case

(-7)×(-2)

= 14

When 45 is added then digits interchange their position

★ First case : adding 45

27 + 45

= 72 (interchanged)

★ Second case : adding 45

-72 + 45

=-27 ( interchanged)