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1. If a number of two digits is k times the sum of its digits, then the number formed by interchanging the digits is the sum of the digits multiplied by :

A. k - 1

B. 11 - k

C. 9 + k

D. 10 - k

Solution:

Let the ten's digit be x and the unit's digit be y
Then, number = 10x + y

\begin{aligned} ∴ 10 x + y = k (x + y) \\ \Rightarrow k = \frac{10 x + y}{x + y} \end{aligned}

Number formed by interchanging the digits = 10y + x
Let, 10y + x = h(x + y)
Then,

\begin{aligned} h = \frac{10 y + x}{x + y} \\ = \frac{11 (x + y) - (10 x + y)}{x + y} \\ = 11 - \frac{10 x + y}{x + y} \\ = 11 - k \end{aligned}

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