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1. In a two-digit number, if it is known that its unit's digits exceeds its ten's digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is :

A. 24

B. 26

C. 42

D. 46

Solution:

Let the ten's digit be x
Then, unit's digit = x + 2
∴ Number :
= 10x + (x + 2)
= 11x + 2
Sum of digits :
= x + (x + 2)
= 2x + 2

\begin{aligned} ∴ (11 x + 2) (2 x + 2) = 144 \\ \Leftrightarrow 22 x^{2} + 26 x - 140 = 0 \\ \Leftrightarrow 11 x^{2} + 13 x - 70 = 0 \\ \Leftrightarrow (x - 2) (11 x + 35) = 0 \\ \Leftrightarrow x = 2 \end{aligned}

Hence, required number:
= 11x + 2
= 11× 2 + 2
= 24

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