In a two-digit number, if it is known that it | vedclass

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Question

(A) Let the ten's digit be $x$ and the unit's digit be $y$. The number is $10x + y$.Given that the unit's digit exceeds the ten's digit by $2$, so $y

Vedclass · Accepted Answer

$24$

Vedclass · Answer

(A) Let the ten's digit be $x$ and the unit's digit be $y$. The number is $10x + y$.Given that the unit's digit exceeds the ten's digit by $2$, so $y = x + 2$.The sum of the digits is $x + y = x + (x + 2) = 2x + 2$.The product of the number and the sum of its digits is $144$, so $(10x + y)(x + y) = 144$.Substituting $y = x + 2$ into the equation:$(10x + x + 2)(x + x + 2) = 144$$(11x + 2)(2x + 2) = 144$$(11x + 2) \cdot 2(x + 1) = 144$$(11x + 2)(x + 1) = 72$$11x^2 + 11x + 2x + 2 = 72$$11x^2 + 13x - 70 = 0$Factoring the quadratic equation:$(11x + 35)(x - 2) = 0$Since $x$ must be a positive integer, $x = 2$.Then $y = x + 2 = 2 + 2 = 4$.The number is $10(2) + 4 = 24$.

In a two-digit number, if it is known that its unit's digit 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

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