Question

If sum of the number $985$ and two other numbers obtained by arranging the digits of $985$ in cyclic order is divided by $111,22$ and $37$ respectively. Find the quotient in each case.

Answer 1

Given number $=985$

Sum of the numbers obtained by cyclic order of the number $985$ $=985+859+598$

Case1:

We know that, if $a,b$ and $c$ are any three digits then $abc+bca+cab$ is exactly divisible by $111$ and the qutient is $a+b+c$.

i.e., $985+859+598$ is exactly divisible by $111$.

$\Rightarrow$ Quotient $=9+8+7=22$

Case2:

In case1 when $985+859+598$ is exactly divisible by $111$, the quotient is $22$

$\Rightarrow$ when $985+859+598$ is divisible by $22$ then the quotient is $111$.

Case3:

In case1 when $985+859+598$ is exactly divisible by $111$, the quotient is $22$

As we know, $111=37\times 3$

$\Rightarrow$ when $985+859+598$ is divisible by $37$ then the quotient is,

$3\times 22=66$