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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.

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Solution

Let $n (a, b)$ be the number.
$n (a, b) = 10 a + b$
Given that, $\frac{n (a, b)}{n (b, a)}$ gives $4$ as quotient and $3$ as remainder.
$n (a, b) = 4 n (b, a) + 3$
$10 a + b = 40 b + 4 a + 3$
$6 a = 39 b + 3$
$2 a = 13 b + 1$
Given that, $n (a, b) = 8 (a + b) + 7$
$10 a + b = 8 a + 8 b + 7$
$2 a = 7 b + 7$
$7 b + 7 = 13 b + 1$
$6 = 6 b$
$b = 1 \Rightarrow 2 a = 13 + 1$
$a = 7$
$71$ is the required number.

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