The sum and difference of LCM and HCF of two numbers are 592 and 518, respectively. If the difference of thenumbers is x and their sum is 296 then the value of x is:

The sum and difference of LCM and HCF of two numbers are $592$ and $518$, respectively. If the difference of the numbers is x and their sum is $296$ then the value of $x$ is:
1). $74$
2). $111$
3). $148$
4). $37$

Let the L.C.M and H.C.F be x and y respectively.  
$x + y = 592$
$x - y = 518$
$x = 555$ & $y = 37$  
Now let the numbers be $37a$ and $37b$ 
$37a +37b = 296$  
$a + b = 8$  
Possible pairs of co-primes, whose sum is $8$ are $(1, 7)$ & $(3, 5)$
Possible pair of numbers are $(37×1, 37×7) = (37, 259) $ 
and $(37×3, 37×5) = (111, 185) $ 
Now $H.C.F × L.C.M. = 555 × 37 = 20535$
Also $111 × 185 = 20535$, while $37 × 259 ≠ 20535$
Hence the required numbers are $111$ and $185$

Difference is $74$

74 is the correct answer

74

We are given the following information:

1. $\text{LCM}+\text{HCF}=592$
2. $\text{LCM}-\text{HCF}=518$
3. Sum of the two numbers = 296
4. Difference of the two numbers = $x$ (to be found)

Step 1: Find LCM and HCF
We solve for LCM and HCF using the given equations:

$\text{LCM}+\text{HCF}=592$

$\text{LCM}-\text{HCF}=518$

Adding both equations:

$2\times \text{LCM}=1110$

$\text{LCM}=\frac{1110}{2}=555$

Subtracting the second equation from the first:

$2\times \text{HCF}=74$

$\text{HCF}=\frac{74}{2}=37$

Step 2: Find the product of the two numbers
We know that the product of the two numbers is given by:

$\text{LCM}\times \text{HCF}=\text{Product of the two numbers}$

$555\times 37=20535$

Step 3: Solve for the numbers
Let the two numbers be $a$ and $b$. We have:

$a+b=296$

$a\times b=20535$

Since we know that:

$a-b=x$

We use the identity:

$(a-b{)}^2=(a+b{)}^2-4ab$

$x^2={296}^2-4(20535)$

$=87616-82140$

$=5476$

$x=\sqrt{5476}=74$

Final Answer:

$\boxed{{\displaystyle x=74}}$

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