Question

# The product of two positive integers is 936. Find the greater number, if the integers are in ratio $13:18.$A.27B.31C.36D.41

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Hint: From the given data we will form two linear equations in two variables(the two integers), and then solve the linear equations by using the substitution method to find the greatest number.

Given data: The product of two positive integers is 936.
The ratio of those integers is $13:18.$
Now, let us assume that the two integers are a and b, where ${\text{b > a}}$
From the given data, we can say that
$ab = 936...........(i) \\ \dfrac{a}{b} = \dfrac{{13}}{{18}}..............(ii) \\ a = \dfrac{{13}}{{18}}b.............(iii) \\$
On substituting the value of ‘a’ in equation (i), we will get
$(\dfrac{{13}}{{18}}b)b = 936 \\ \Rightarrow {b^2} = 936\left( {\dfrac{{18}}{{13}}} \right) \\ On{\text{ }}simplifying{\text{ }}we{\text{ }}get, \\ \Rightarrow {b^2} = 72(18) \\ \Rightarrow {b^2} = 1296 \\ On{\text{ }}taking{\text{ }}square{\text{ }}root{\text{ }}we{\text{ }}get,{\text{ }} \\ \Rightarrow b = \pm 36 \\$
But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘b’ i.e.
$b = 36$
Now, putting the value of ‘b’ in equation (iii)
$a = \dfrac{{13}}{{18}}(36) \\ \Rightarrow a = 13(2) \\ \Rightarrow a = 26 \\$
From the assumption we made it is clear the ‘b’ is the greater integer and ${\text{b = 36}}{\text{.}}$
Hence, option (C) is correct.

Note: While solving for ‘a’ and ‘b’ we can also substitute the value of ‘b’ as is shown below
$ab = 936, \\ \Rightarrow \dfrac{a}{b} = \dfrac{{13}}{{18}} \\ \Rightarrow a = \dfrac{{13}}{{18}}b \\ \Rightarrow b = \dfrac{{18}}{{13}}a..........(iv) \\$
On substituting the value of ‘b’ in equation (i), we will get
${\text{a(}}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{a) = 936}} \\ \Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 936}}\left( {\dfrac{{{\text{13}}}}{{{\text{18}}}}} \right) \\ {\text{On simplification we get,}} \\ \Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 52(13)}} \\ \Rightarrow {{\text{a}}^{\text{2}}}{\text{ = 676}} \\ {\text{On taking square root, we get,}} \\$
$\Rightarrow$ $a$ =$\pm$ $26$

But it is given that ‘a’ and ‘b’ are positive integers, giving us the value of ‘a’ i.e.
${\text{a = 26}}$
Now, putting the value of ‘a’ in equation (iv)
${\text{b = }}\dfrac{{{\text{18}}}}{{{\text{13}}}}{\text{(26)}} \\ \Rightarrow {\text{b = 18(2)}} \\ \Rightarrow {\text{b = 36}} \\$
Additional information: Square root of any number always gives us the answer in pairs either it is of a real number or complex number. Here we also get a pair of values of ‘b’ but it is mentioned that the integers are positive, so we proceed further with the positive value of ‘b’.