Question

A two digit number is such that the product of its digits is 16. When 54 is subtracted from the
number, the digits are interchanged. Find the number.
​

Answer 1

Answer:

(x)*(y)=16

(10x+y)-54=(10y+x)

9x-54=9y

9(x-y)=54

x-y=6

x=6+y

substituting in (1)

we get y=2

x=8

Let the two digit number be 10x + y

Given : product of its digits(xy) = 16

xy = 16...................(1)

When 54 is subtracted from the number, the digits interchange their places

10x + y - 54 = 10y + x

10x + y - 10y - x = 54

9x - 9y = 54

9(x - y) = 54

x - y = 54/9

x - y = 6

x = 6 + y……………….(2)

Put this value of x in eq 1.

xy = 16

(6 + y)y = 16

6y + y² = 16

y² + 6y - 16 = 0

y² + 8y - 2y - 16 = 0

[By middle term splitting]

y(y + 8) - 2(y + 8) = 0

(y - 2 ) ( y + 8) = 0

(y - 2 ) = 0 or ( y + 8) = 0

y = 2 or y = - 8

Since, a digit can't be negative, so y ≠ - 8.

Therefore , y = 2

Put this value of y in eq 1,

xy =16

x× 2 = 16

x = 16/2 = 8

x = 8

Required number = 10x + y

= 10(8) + 2

= 80 + 2

Required number = 82

Hence, the Required two digit number is 82.

Answer 2

Answer:

$\large{\underline{\boxed{\sf Number = 82}}}$

Step-by-step explanation:

Let the digit at unit's place be x and the digit at ten's place be y.

∴ Original number = 10x + y

It is given that ;

Product of its digits is 16.

⇒ xy = 16

According to the question ;

⇒ 10x + y - 54 = 10y + x

⇒ 10x - x + y - 10y = 54

⇒ 9x - 9y = 54

⇒ 9(x - y) = 54

⇒ x - y = $\sf \dfrac{54}{9}$

⇒ x - y = 6... (i)

Squaring both sides ;

⇒ (x - y)² = 6²

⇒ x² + y² - 2 xy = 36

⇒ x² + y² - 2 * 16 = 36

⇒ x² + y² = 36 + 32

⇒ x² + y² = 68

Adding (2xy) both sides ;

⇒ x² + y² + 2xy = 68 + 2 * 16

⇒ (x + y)² = 100

⇒ x + y = ±√100

⇒ x + y = 10 ... (ii) [Taking positive value]

_______________________

On adding equation (i) and (ii),

⇒ x - y + x + y = 6 + 10

⇒ 2x = 16

⇒ x = $\sf \dfrac{16}{2}$

⇒ x = 8

Substituting the value of x in (i)

⇒ x - y = 6

⇒ 8 - y = 6

⇒ y = 8 - 6

⇒ y = 2

_______________________

∴ Original number = 10x + y

= 10 * 8 + 2

= 80 + 2

= 82

Therefore, the original number is 82.