Question

a two digit number such that its product of digits is 16. when 54 is subtracted from the number, the digits are interchanged.find the number.

Answer 1

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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Answer 2

Answer:

Hence, the two-digit number is 82, and the number with interchanged digits is 28.

Step-by-step explanation:

Given - 2 digit number, with the digits' product as 16.
To find - the number
Let, x be 1 digit, and y be the other digit.
Now, we can represent the number as $10x+y$.
Also, we know that $x \times y = 16$
Equations -
$x \times y = 16$                                     → equation 1
$(10x + y) - 54 = 10y + x$               → equation 2
Solving the 2 equations as follows, we get
equation 2 → $10x - x - 54 = 10y - y$
                      $9x - 54 = 9y$
                      $x - 6 = y$
substituting equation 1 in 2, we get
$x - 6 = \frac{16}{x}$
solving further, this equation can be simplified as
$x^{2} - 6x = 16$
$(x - 8)(x+2) = 0$
so, x is either 8 or -2.
because we're considering positive integers, we take $x = 8$
therefore, by substituting the value of x in equation 1, we get $y = 2$
thus, the first number is 82, and the interchanged number is 28.

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