Question

LCM of two numbers is 45 times their HCF. If one of the numbers is 125 and the sum of their HCF and LCM is 1150, find the other number.

Answer 1

Hint: First we need to assign certain variables to LCM, HCF and the number. Now, create linear equations according to the conditions mentioned. Solve the equation and find the value of HCF and LCM. Use the principle, the product of two numbers is equal to the product of HCF and LCM and further solve it to find the required result which is the other number along with 125.

Complete step by step answer:
Let us consider LCM as $x$, HCF as $y$ and the number which is to be found is $\text{z}$, the other given number is 125.
According to the conditions,
We know,
LCM of two numbers is 45 times their HCF, we get
$x$= 45 x$y$
$x$= $45y$...................(1)
Now, the other condition given is sum of the HCF and LCM of those two numbers is equal to 1150, we get
$x+y$ = 1150 .....................(2)
Now, let us substitute, equation (1) in equation (2), we get
$45y$ + $y$ = 1150
         $46y$ = 1150
              $y$ = $\dfrac{1150}{46}$
                   = 25
Therefore, we got the HCF as 25, now let us find out how much will be LCM. In order to do that, let us substitute the HCF, $y$= 25 value in equation (1), we get
$x$= 45 x 25
    = 1125
So, we got LCM as 1125.
We know, that the product of two number is equal to the product of HCF and LCM, therefore we get
Product of two numbers = Product of HCF and LCM
                              $\text{z}$ x 125 = HCF x LCM
$\text{z}$ x 125 = 25 x 1125.
In the above equation, divide by 125 on both the sides of the equation, we get
$\dfrac{\text{z}\times \text{125}}{125}=\dfrac{25\times 1125}{125}$
           $\text{z}$$=\dfrac{28125}{125}$
                = 225

Therefore, the other number along with the 125 is number 225 whose HCF is 25 and LCM is 1125.

Note: HCF is Highest Common Factor and LCM is Least Common Multiple, the above equations that we used are in the form of two variable linear equations. General form, ax + by = c, where a, b and c are coefficients and x and y are the deciding variables.