# Sum of two numbers is 407. The sum and difference of their LCM and HCF are 925 and 851 respectively. The difference between the two numbers is?

(a) 518

(b) 185

(c) 158

(d) 175

Answer

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**Hint:**Assume the two numbers as x and y, form the first equation by taking the sum of x and y and equating it with 407. Now, form two equations related to LCM and HCF using the given information and solve for their values. Use the formula LCM of two numbers = $\dfrac{x\times y}{\text{HCF}}$ and equate it with the found value of LCM. Form a second relation between x and y, finally use the formula $\left| x-y \right|=\sqrt{{{\left( x+y \right)}^{2}}-4xy}$ to get the answer.

**Complete step by step answer:**

Let us assume the two numbers as x and y. It is given that the sum of these numbers is 407, so we get,

$\Rightarrow x+y=407...........\left( i \right)$

Now, we have been given that the sum and difference of the LCM and HCF of these numbers are 925 and 851 respectively, so we have two relations given as: -

$\Rightarrow $ LCM + HCF = 925

$\Rightarrow $ LCM – HCF = 851

Solving above two equations using the elimination method we get LCM = 888 and HCF = 37. We know that the LCM of two unknown numbers x and y is given as $\dfrac{x\times y}{\text{HCF}}$, so we get,

$\begin{align}

& \Rightarrow 888=\dfrac{x\times y}{37} \\

& \Rightarrow xy=37\times 888 \\

& \Rightarrow xy=32856.........\left( ii \right) \\

\end{align}$

Now, we can write ${{\left( x-y \right)}^{2}}={{\left( x+y \right)}^{2}}-4xy$ so we taking square root both the sides we get,

$\Rightarrow \left| x-y \right|=\sqrt{{{\left( x+y \right)}^{2}}-4xy}$

Substituting the obtained values from equations (i) and (ii) in the above relation we get,

$\begin{align}

& \Rightarrow \left| x-y \right|=\sqrt{{{\left( 407 \right)}^{2}}-4\times 32856} \\

& \Rightarrow \left| x-y \right|=\sqrt{165649-131424} \\

& \Rightarrow \left| x-y \right|=\sqrt{34225} \\

& \therefore \left| x-y \right|=185 \\

\end{align}$

**So, the correct answer is “Option b”.**

**Note:**You may think why we have taken the modulus sign in the L.H.S while solving for (x – y), the simple reason is that we don’t know which of the numbers x or y is greater so on taking modulus we will always get the positive result. Remember the formula of the LCM of two unknown numbers to solve the above question.

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