Question

19. Th e sum of a two-digit number and the number formed by interchanging the digits is 132. Of 12 is added to the number, the new number becomes 5 time
the sum of the digits. Find the number. the sum of digits. When D

Answer 1

We can write the unknown number as $10x+y$, where both $x,y$ are in the set $\{1,2,\ldots,8,9\}$. (Neither can be 0)

Interchanging the digits makes the number $10y+x$. So

$10x+y+10y+x=11(x+y)=132\implies x+y=12$

Adding 12 to the number makes it 5 times the sum of the number's digits, which means

$10x+y+12=5(x+y)\iff 5x-4y=-12$

Now we can solve the system,

$\begin{cases}x+y=12,5x-4y=-12\end{cases}\implies x=4,y=8$

so the original number is 48.