Question

If we divide the unknown two-digit number by the number consisting of the same digits written in the reverse order, we get $4$ as a quotient and $3$ as a remainder. Now if we divide the required number by the sum of its digits, we get $8$ as a quotient and $7$ as a remainder. Find the number.

Answer 1

Let $n(a,b)$ be the number.

$n(a,b)=10a+b$

Given that, $\frac{n(a,b)}{n(b,a)}$ gives $4$ as quotient and $3$ as remainder.

$n(a,b)=4n(b,a)+3$

$10a+b=40b+4a+3$

$6a=39b+3$

$2a=13b+1$

Given that, $n(a,b)=8(a+b)+7$

$10a+b=8a+8b+7$

$2a=7b+7$

$7b+7=13b+1$

$6=6b$

$b=1\Rightarrow 2a=13+1$

$a=7$

$71$ is the required number.