The sum 52

The sum of digits of a two digits number is 10. On interchanging the digit, the number obtained is 54 less than the original number. What is the original number?

Correct answer:

x =  82

Step-by-step explanation:

$$\begin{gathered} x=ab=10\cdot a+b \\ a+b=10 \\ a>0 \\ 10b+a=10a+b-54 \\ a+b=10 \\ 10\cdot b+a=10\cdot a+b-54 \\ a+b=10 \\ 9a-9b=54 \\ a+b=10 \\ Row2-\frac{1}{9}\cdot Row1\to Row2 \\ 9a-9b=54 \\ 2b=4 \\ b=\frac{4}{2}=2 \\ a=\frac{54+9b}{9}=\frac{54+9\cdot 2}{9}=8 \\ a=8 \\ b=2 \\ x=10\cdot a+b=10\cdot 8+2=82 \\ \text{ Verifying Solution: } \\ y=10\cdot b+a=10\cdot 2+8=28 \\ \Delta =x-y=82-28=54 \end{gathered}$$

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