A two-digit number divided by the sum of its digits gives 7 and leaves no remainder. If the digits are interchanged, the new number is 36 less than the original. Find the original number.

Let the digits be $x$ (tens) and $y$ (units). $10x+y=7(x+y)$ gives $3x=6y$, so $x=2y$. Also $(10x+y)-(10y+x)=36$ gives $9(x-y)=36$, so $x-y=4$. Then $2y-y=4,y=4$ and $x=8$, giving 84. Check: 84 / 12 = 7 and 84 - 48 = 36. 21 and 63 also divide by their digit sums to give 7, but they fail the second condition (21 - 12 = 9 and 63 - 36 = 27); 48 is the reversed number.