Question

A two-digit number is such that the product of the 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

(x)*(y)=16..........(1)

(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