Question

The product of the digits of a two digit positive number is 24. If 18 is added to the
number then the digits of the number are interchanged. Find the number.​

Answer 1

☯ Let the ten's digit of the number be x.

Given that,

• The product of the digits of a two digit positive number is 24.

$:\implies\sf Unit's \ digit = \dfrac{24}{x}$

And,

$:\implies\sf Two \: digit \: number = 10x + \dfrac{24}{x}$

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━

⠀⠀⠀⠀⠀⠀

⠀⠀⠀$\boxed{\bf{\mid{\overline{\underline{\bigstar\: According\: to \: the \: Question :}}}}\mid}$

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• If 18 is added to the number then the digits of the number are interchanged.

$:\implies\sf 10 \times \dfrac{24}{x} + x$

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Now,

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$:\implies\sf 10x + \dfrac{24}{x} + 18 = 10 \times \dfrac{24}{x} + x \\\\\\:\implies\sf 10x^2 + 24 + 18x = 240 + x^2 \\\\\\:\implies\sf 10x^2 - x^2 + 18x = 240 - 24 \\\\\\:\implies\sf 9x^2 + 18x = 216 \\\\\\:\implies\sf 9x^2 + 18x - 216 = 0 \\\\\\:\implies\sf 9 \Big(9x^2 + 18x - 216 \Big) = 0 \\\\\\:\implies\sf x^2 + 2x - 24 = 0 \\\\\\:\implies\sf x^2 + 6x - 4x - 24 = 0 \\\\\\:\implies\sf x(x + 6) -4 (x + 6) = 0 \\\\\\:\implies\sf (x - 4) (x + 6) = 0 \\\\\\:\implies\sf x = 4 \: \: \& \ \ x = -6$

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• Ignoring negative value, x is 4.

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Therefore,

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$:\implies\sf The \: number = 10x + \dfrac{24}{x}\\\\\\:\implies\sf 10 (4) + \cancel\dfrac{24}{4} \\\\\\:\implies\sf 40 + 6 \\\\\\:\implies{\underline{\boxed{\frak{\pink{46}}}}} \ \bigstar$

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$\therefore\:{\underline{\sf{Hence,\ \ required \ two \ digit \ number \:is\: \frak{46}.}}}$

Answer 2

Step-by-step explanation:

$\frak Given = \begin{cases} &\sf{The\ product\ of\ the\ digits\ of\ a\ two\ positive\ number\ is\ 24.} \\ &\sf{And\ number\ 18\ is\ added\ to\ the\ number\ then\ the\ digits\ of\ the\ number\ are\ interchanged.} \end{cases}$

To find:- We have to find the number?

☯️ Let the digit at ten's place be x.

___________________

$\frak{\underline{\underline{\dag So\ here:-}}}$

$\sf : \implies {The,\ digit\ at\ one's\ place\ =\ \dfrac{24}{x}} \\ \\ \sf : \implies {Original\ number\ =\ 10x\ +\ \dfrac{24}{x}} \\ \\ \sf : \implies {\underline{On\ interchanging\ the\ numbers:-}} \\ \\ \sf : \implies {New\ number\ =\ \dfrac{240}{x}\ +\ x}$

___________________

$\frak{\underline{\underline{\dag According\ to\ the\ question:-}}}$

$\sf : \implies {10x\ +\ \dfrac{24}{x}\ +\ 18\ =\ \dfrac{240}{x}\ +\ x} \\ \\ \sf : \implies {10x²\ +\ 24\ +\ 18x\ =\ 240\ +\ x²} \\ \\ \sf : \implies {9x²\ +\ 18x\ -\ 216\ =\ 0} \\ \\ \sf : \implies {x²\ +\ 2x\ -\ 24\ =\ 0} \\ \\ \sf : \implies {(x\ +\ 6)\ (x\ -\ 4)\ =\ 0} \\ \\ \sf : \implies {\purple{\underline{\boxed{\bf x\ =\ -6\ or\ 4.}}}}\bigstar$

● But here x can't be negative, so x = 4.

☯️ So here:-

$\sf : \implies {2\ digits\ number\ =\ \dfrac{10x\ +\ 24}{x}} \\ \\ \sf : \implies {\dfrac{10\ ×\ 4\ +\ 24}{4}} \\ \\ \sf : \implies {40\ +\ 6} \\ \\ \sf : \implies {\purple{\underline{\boxed{\bf 46.}}}}\bigstar$

Hence:-

$\sf \therefore {\underline{The\ number\ is\ 46.}}$