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.