Question

The sum of two number is 916 .if 27 is their HCF then .
the number are​

Answer 1

$\large\underline\blue{\bold{Given \:  Question \tt (correct \: statement) :-  }}$

The sum of two number is 216. If 27 is their HCF, then the numbers are

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$\tt\:\huge \red{AηsωeR} ✍$

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$\begin{gathered}\begin{gathered}\bf Given - \begin{cases} &\tt{sum \: of \: two \: numbers \: is \: 216} \\ &\tt{HCF \: = 27} \end{cases}\end{gathered}\end{gathered}$

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$\begin{gathered}\begin{gathered}\bf To \: Find :- \begin{cases} &\tt{the \: two \: numbers} \end{cases}\end{gathered}\end{gathered}$

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Concept Used:-

• If HCF of two numbers X & Y is k, then the numbers can be written as ka & kb respectively such that a & b are co-prime to each other.

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$\large\underline\purple{\bold{Solution :- }}$

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$\begin{gathered}\begin{gathered}\bf Let - \begin{cases} &\tt{first \: number \: be \: a} \\ &\tt{second \: number \: be \: b} \end{cases}\end{gathered}\end{gathered}$

$\tt \: Now, \: it \: is \: given \: that \: HCF(a, \: b) \: = \: 27$

$\tt \therefore \: a \: = \: 27x \: \: \: and \: \: b \: = \: 27y$

$\tt \: where \: x \: and \: y \: are \: co - prime \: to \: each \: other.$

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$\tt \: \begin{gathered}\red{ \tt \: According \: to \: statement}\end{gathered}$

$\tt\implies \:a \: + \: b \: = \: 216$

$\tt\implies \:27x \: + \: 27y \: = \: 216$

$\tt\implies \:27(x + y) = 216$

$\tt\implies \:x \: + \: y \: = \: 8$

$\tt \: So, \: we \: have \: two \: conditions \: with \: us \: now. \\ \tt \: (a + b) \: is \: 8 \: and \: a \: and \: b \: are \: co-prime. \: \: \:$

$\tt \implies \: Possible \: pairs \: of \: a \: and \: b \: are(1,7), \: (3,5)</p><p></p><p>$

$\tt \: Thus \: possible \: pairs \: of \: the \: numbers \: are$

$\tt \large\boxed{ \tt(27, \: 189), \: (81, \: 135)}$

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Note :-

☆ Statement is wrong as 916 is not divisible by 27. It must be a multiple of 27, so might it be 216 or 918

☆ If it is 918, then x + y = 34

☆ so possible pairs are (3, 31), (5, 29), (11, 23), (17, 17)