Given:

x + y = 42
xy = 437
Find:

|x - y|
Substitute the first equation into the expression for |x - y| to get:

|x - y| = √((x + y)^2 - 4xy)
Substitute the given values for x + y and xy into this expression to get:

|x - y| = √((42^2) - 4(437)) = √(1764 - 1748) = √(16) = 4
Therefore, the absolute difference between the numbers is 4.

To find the number of numbers between 100 and 600 that are divisible by 2, 3, and 7 simultaneously, we can first find the least common multiple (LCM) of 2, 3, and 7. The LCM of 2, 3, and 7 is 42, which means that any number that is divisible by 2, 3, and 7 must also be divisible by 42.
We can then count the number of multiples of 42 between 100 and 600. The first multiple of 42 greater than or equal to 100 is 105, and the last multiple of 42 less than or equal to 600 is 588. The number of multiples of 42 between these two numbers is:
(588 - 105) / 42 + 1 = 12
Therefore, there are 12 numbers between 100 and 600 that are divisible by 2, 3, and 7 simultaneously.

To maintain the same expenditure on rice when the price increases from Rs. 6 per kg to Rs. 8 per kg, a person must reduce their consumption of sugar by 25%.

To see this, suppose that a person was originally spending X rupees on rice, and consuming Y kg of sugar. Then, the original cost of the sugar can be expressed as 6X rupees/Y kg.

When the price of rice increases to Rs. 8 per kg, the person must reduce their consumption of sugar in order to maintain the same expenditure on rice. Let's say they reduce their consumption of sugar to Y - Z kg, where Z is the amount of sugar they need to reduce.

The cost of the sugar after the reduction is (8X rupees)/(Y - Z kg). Setting this equal to the original cost of the sugar, we have:

(8X rupees)/(Y - Z kg) = 6X rupees/Y kg

Solving for Z, we find that Z = (3/4)Y. In other words, the person must reduce their consumption of sugar by 25% in order to maintain the same expenditure on rice when the price increases from Rs. 6 per kg to Rs. 8 per kg 

Letting the positive number be x and then expressing the first condition in terms of x, we get:

2/3 x = 16/216 * 1/x

We can then multiply both sides by 216 and simplify to get:

x^2 = 16/144

Taking the square root of both sides gives:

x = sqrt(16/144)

Simplifying this expression gives:

x = sqrt(1/9)

Finally, since the number must be positive, we take the positive square root, which gives us:

x = 1/3 or 4/12

Therefore, the positive number that satisfies the given conditions is 1/3. 

 When multiplying numbers, the unit digit of the final product is determined by the product of the unit digits of the individual numbers. 

For example, if we multiply 148 by 163 by 236, the unit digit of the final product will be the product of the unit digits of each of these numbers, which is 8 * 3 * 6 = 24. 

Since 4 is the unit digit of 24, the unit digit of the final product will be 4.

 In this problem, we are given that 12.5% of a group of books are non-fiction, which can also be written as 1/8th of the books. We are also given that 1/6th of the books are fiction. We are trying to determine the total number of books in the group.

Since 1/6th of the books are fiction and 1/8th are non-fiction, we know that the total number of books must be a multiple of both 6 and 8. This means that the total number of books must be divisible by their least common multiple, which is 24.

We are also told that the total number of books must be between 75 and 100. This means that the possible values for the total number of books are 96, 72, 48, etc.

Of these possible values, only 96 fulfills the criteria of being divisible by 24 and being between 75 and 100. Therefore, we can conclude that the total number of books is 96

To find the values of x and y, we can use the information provided to create a system of equations and solve for x and y.

The first equation is given: x + y = 42
The second equation is given: xy = 437

To find the difference between x and y, we can use the equation x - y = sqrt[(x + y)^2 - 4xy], which is also given.

Substituting the values from the first two equations into the third equation, we get: x - y = sqrt[(42)^2 - 4 x 437] = sqrt[1764 - 1748] = sqrt[16] = 4.

From this equation, we can see that the difference between x and y is 4.
To find the values of x and y, we can set up the following system of equations:

x + y = 42
x - y = 4

Solving this system of equations using substitution, we get:

x = 23 and y = 19

Therefore, the values of x and y are 23 and 19, respectively.

The sum of these four integers is 1290, so we can write the equation:
x + (x+1) + (x+2) + (x+3) = 1290

Simplifying this equation, we get:
4x + 6 = 1290

Subtracting 6 from both sides, we get:
4x = 1284

Dividing both sides by 4, we get:
x = 321

So, the four consecutive integers are 321, 322, 323, and 324.
The greatest of these integers is 324.
Therefore, the answer to the question is 324.

We are given two numbers, A and B, such that A > B. The product of these numbers is 266 and the value of B is equal to 266 divided by A. We are also told that the difference between A and B is 5. Using this information, we can set up the following system of equations:

Equation 1: AB = 266
Equation 2: B = 266/A
Substituting Equation 2 into Equation 1, we get:

(266/A)A = 266

This simplifies to:

A^2 - 5A + 266 = 0

(A – 19)(A – 14) = 0
Therefore , A = 19 or A = 14
Hence, bigger number = A = 19

Let numbers be A and B

.: (A-B)^2=9

..A^2-2AB+ B^2 = 9

As, A^2+B^2=225

225-2AB = 9

AB = 108 product of the two numbers