Question

A two digits number is such that the product of the digits is 16 when 54 is subtracted from the number , the digits inter changed find the number?

Answer 1

Let us assume, x and y are the two digits of the two-digit number.

Therefore, the number = 10x + y and reversed number = 10y + x

Given:

x * y = 16 
x = 16/y -------------1

Also given:

10x + y - 54 = 10y + x
9x - 9y = 54
x - y = 6 ------------2

Substitute the value of x from equation 1 in equation 2

16/y - y = 6
16 - y^2 = 6y
y^2 + 6y - 16 = 0
(y + 8) (y -2 ) = 0
y + 8 = 0 or y - 2 = 0
y = -8 or y = 2

Since y can not be negative, y = 2

Therefore, x =16/y = 16/2 = 8

Therefore, the two digit number =  82

Answer 2

SOLUTION:

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.

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