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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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