Question

A three digit number is equal to $17$ times the sum of its digits. If $198$ is added to the number, the digits are interchanged The addition of first and third digit is $1$ less than middle digit. Find the number

Answer 1

Let the number be $abc$

So number is $100a+10b+c$

$\Rightarrow 100a+10b+c=17(a+b+c)$

$\Rightarrow 83a=7b+16c$ .......(i)

$198+100a+10b+c=a+10b+100c$

Or $198+99a=99c$

Hence $c-a=2\Rightarrow c=a+2$ .....(ii)

Also, given $a+c=b-1$

Now as $a+c=b-1$ and $c=a+2$

$\Rightarrow a+a+2=b-1$

$\Rightarrow b=2a+3$

now in eq(i)

substitute the values of b and c

$83a=7b+16c$

$\Rightarrow 83a=7(2a+3)+16(a+2)$

$\Rightarrow 53a=21+32$

$\Rightarrow a=1$

$\Rightarrow c=a+2=3$

$\Rightarrow b=2a+3=5$

So the number is $153$