A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.

Explanation

Let the tens and units digits of the required number be x and y respectively. Then,
`xy=18impliesy=(18)/(x)" "…(i)`
And, `(10x+y)-63=10y+x`
`implies" "9x-9y=63impliesx-y=7" "…(ii)`
Putting `y=(18)/(x)` from (i) into (ii), we get
`x-(18)/(x)=7`
`impliesx^(2)-18=7ximpliesx^(2)-7x-18=0`
`impliesx^(2)-9x+2x-18=0impliesx(x-9)+2(x-9)=0`
`implies(x-9)(x+2)=0impliesx-9=0" or "x+2=0`
`impliesx=9" or "x=-2`
`impliesx=9" "[because" a digit cannot be negative"].`
Putting x=9 in (i), we get y=2.
Thus, the tens digit is 9 and the units digit is 2.
Hence, the required number is 92.