Question

Find the largest number thqt exactly divide 513,783,1107

Answer 1

it is 27
hope this might help you

Answer 2

$\underline{\underline{\textbf{\Huge{Question:}}}}$
Find the largest number that exactly divide 513,783,1107



$\underline{\underline{\huge{\textsf{Concept\: Behind: }}}}$

$\textsf{No. that will divide \implies Divisor}$

$\therefore$ Divisor has to be found.

The Given Numbers in the question are the Dividends.

$\bigstar \: \textsf{The Greatest No. that will divide x, y and z is \textbf{H.C.F[x, y, z]}}$

$\textsf{H.C.F}$ is also called as $\textbf{Greatest Common Divisor (G.C.D)}$


$\underline{\underline{\LARGE{\textbf{Solution:}}}}$

$\underline{\large{\textit{Step-1:}}}$
Express the Given Numbers as $\textit{Product of Prime factors}$ using $\textbf{Prime factorisation}$

$\begin{array}{r|l} 3&513 \\ \cline{2-2} 3 & 171\\ \cline{2-2}3 & 57\\ \cline{2-2} 19& 19\\\cline{2-2}&1\end{array} \quad\quad \begin{array}{r|l} 3&783 \\ \cline{2-2} 3 & 261\\ \cline{2-2} 3 & 87\\ \cline{2-2} 29& 29\\\cline{2-2} &1\end{array} \quad \quad \begin{array}{r|l} 3&1107 \\ \cline{2-2} 3 & 369 \\ \cline{2-2} 3 & 123 \\ \cline{2-2} 41& 41 \\\cline{2-2} &1\end{array}$


$513\: = 3 \times 3 \times 3 \times 19$
$783\: = 3 \times 3 \times 3 \times 29$
$1107 = 3 \times 3 \times 3 \times 41$


$\underline{\large{\textit{Step-2:}}}$
Note the $\textsf{common prime factors}$

Common Prime factors = 3, 3, 3


$\underline{\large{\textit{Step-3:}}}$
Multiply all the common prime factors to get the $\textbf{HCF}$ of the given numbers.

$\implies$ Required HCF = $3\times3\times3= 27$

$\blacksquare \: \textsf{The greatest No. that will divide 513, 783 and 1107 is \underline{\large{\textbf{27}}}}$


$\underline{\Large{\textsf{Verification:}}}$

$513\: = 19(27)+0$
$783\: = 29(27)+0$
$1107 = 41(27)+0$