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72 is the original number
72 - 27 = 45
(7+2)*(7-2) = 9*5 = 45
Here's how to do it via algebra:
Choose x as ten's & y as the one's digit of your number.
So your original number is 10x+y
When you reverse the digits of the number the new number (10y+x) is 45 less than the original so:
10y+x =10x+y-45
10y-10y+x-x =10x-x+y-10y-45+45 (rearrange)
9x-9y=45 (you can simplify by dividing by 9)
x-y =5 [1st equation]
Next the sum of the digits times the difference of the digits is 45 so we have this equation:
(x+y)*(x-y)=45 [2nd equation]
Now we have 2 equations with 2 unknowns. From the first equation : (x-y)= 5 we can substitute that value into the 2nd equation, so:
(x+y)*(5)=45
(x+y) =9 [3rd equation]
Now using the 3rd equation and the 1st equation we can solve for x & y.
x-y= 5
x+y=9
--------------- (add the 2 equations)
2x =14
x= 7
Substitute 7 into either equation above to solve for y.
(7) + y=9
y= 9 - 7 = 2
So the tens digit is 7 & ones digit is 2 hence 72 is the original number.
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