A two - digit number is such that product of its digits is 14. If 45 is added to the number, the digits interchange their places. Find the number.

Options

A. `x=2,y=7`
B. `x=3,y=7`
C. `x=2,y=6`
D. `x=4,y=7`

Answer

A

Explanation

Let the ten's and until's digits of the required number be x and y respectively. Then, xy = 14.
Required number = `(10 x + y )`
Number obtained on reversing its digits = ` ( 10 y + x )`
` therefore ( 10 x + y ) + 45 = ( 10 y + x )`
`rArr 9 ( y - x ) = 45 rArr y - x = 5 " " `... (i)
Now, ` (y- x ) ^(2) - ( y - x ) ^(2) = 4 xy `
` rArr (y + x ) = sqrt (( y - x ) ^(2) + 4x y ) = sqrt ( 25+ 4 xx 14)=sqrt ( 81)`
`rArr y + x =9 " " `... (ii) `" " [because ` digits are never negative `]`.
On addint (i) and (ii), we get
` 2y = 14 rArr y = 7`.
Putting `y = 7 ` in (ii), we get
` 7 + x = 9 rArr x = 9 - 7 = 2. `
`therefore x = 2 and y = 7 `
Hence, the required number is 27.